nonlinear hyperbolic systems and transport equations in mathematical biology t. hillen1 and kp

26

Upload: others

Post on 20-Feb-2022

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

����������� ���������� �������������� �"!#%$&�'�(��)+*,�-�(���.���/�#!1032 4��5!.����������76 �5!.8�%$&�5!.���9�(�;:<�������/=/�

>@?BADCFEFEHGJI KMLNIPORQS?UT(? ADLVOWGXEYGXZ\[

]_^W`ba ZcLVd afehg/ikjMlnm�oqpVrVots�rvu�wJrkjyxzl5{Ywqs�|qs�w~}k�"l5wq{W�zpfmnjbsoq��m��zpk|�oqpVr�u�w\�X�zpV|���pVrV����X�zr"}Voqxzlb�-�Y{Hs�w~u���jyxzx.�"�Uw~xzwq|q�@mnw��qjbs�mnjb�"s�otmnj��.wq�"}Vxzotm��zw~p�rV�XpVoqu@�z�bl��ots�j�s�jyoq��m��Uw~pk�rV���-}Vl��Uw~p_u�wXrVjyxzlb�9g(ikjb���Noqlnj�w~p��5s�w\�/pV�zoqpvu�wqm��zw~p-�J�/iV�z�i_�zlh�iVots�oq��mnjbs��U�bjyrv�X�w~pkj�l���pk|~xUj��"otsoqu�jbmnjbs��Fr"���.}"l��Uw~p���wXjb�@�b�Ujyp�m��/��p��"ots�m����b}Vxzots({Ywqs�u�w\�X��pk|@�"oq��mnj��s��zo�oqpVrD�Fl�xz�zu�j(u�w~xzrN�Woqu�wXjb�"otjq�Jrkjbm�oq�zxUjyr���pk{Hwqsu�otm��Uw~p�w~p@��pVrV�U�J�zrV}Voqx�u�w��qjyu�jypfm�#jyi"oy�J�Uwqs��zl�oy�toq�zxzot�"xzj��Fln�.jbjyr-��s�}Vp3m��zu�jylb��m�}ks�p�oqpk|~xUj�rV�zl�mns��U�"}km��zw~pVl�����{�l�}V�i�zpk{Ywqs�u�otm��Uw~pD�zl/jyp�mnjbs�jyr��zp�mnw�u�wXrkjyx�l({Ywqs��.wq�"}"xzotm��Uw~pVlb�"m�ikjyp,s�jyoq��m��Uw~pJ��mns�oqpVl��#wqs�mjy �}Votm��zw~pVl5wqshi��X�#jbs��#w~x��z�hjy �}"otm��Uw~pVl��YmnjyxUjb|qs�ot�"i�jy X}Votm��Uw~pVlb�krVoqu��#jyr@�5o��qjMjy �}"oc�m��Uw~pVl�%s�jyl�}VxUmb�g(ikj�|qw~oqx"wq{-m�iV��l s�jb�J�Ujb�¡��l mnw¢�Vs�jylnjyp�m�lnw~u�j��"oql����(ot�V�"xz���botm��Uw~pVl9wq{-mns�oqp"ln�.wqs�m

jy �}Votm��zw~pVl(oqpVr�i����.jbs��.w~xz���/ln�Jlnmnjyu�l(oqp"rvmnw��zxzxz}Vl�mns�otmnj�m�iVj���w~pVpkjy��m��Uw~pVl(�#jbm��%jbjypmns�oqpVln�.wqs�mvjy X}Votm��Uw~pVlb�Mi����.jbs��.w~xz���,u�wJrkjyxzlb��oqpVr£s�jyoq��m��zw~pJ�¤rV���-}Vl��Uw~pRjy X}Votm��Uw~pVly�¥ �V�Nxz�Ujyr�mnw��ikjyu�w~lnjyp"l��Um��U�qj�u�w\�qjyu�jypfm@�F�ikjyu�wqm�oc¦k�zl��{F}VpV��m��Uw~p"oqxPjylnm���u�otmnjyl�{Hwqsm�ikj5pVw~pVxz�zpkjyots�Um��Ujyl-�zpMm�iVj%�bxzoql�l��z�boqx��ikjyu�wqm�oc¦k�zlWu�wXrVjyxN�F§Botm�xzot¨��¤©MjyxzxUjbsn�«ªJjb|qjyxH���boqp�#j�rVjbs��U�qjyr-�k�"oql�jyr�w~p�m�ikj ��pVrV�U�J�zrV}VoqxV�#jyi"oy�J�Uwqs(wq{B��jyxzxzl(oqpVr,otmnmns�oq��m�oqpfm�ly�¬ i"�zxUj9s�jb�.wqs�m��zpV|%w~p mns�oy�qjyxz��pk|�{Hs�w~pfm-�Vs�wq�"xUjyu�lb�\u�wXrVjyxzl.�/�Um�iM X}V�Ujyl���jyp�m�lnm�otmnjylb�

oqpVr�jb�"�zrVjyu��z�5u�wXrVjyxz�zpk|��zp¢�"oql�l���pk|k�~oMrkjbm�oq�zxUjyr�s�jb�X�Ujb�­��lB|~�U�qjyp�w~p�m��%w u�jbm�ikwXr"lwq{#s�jyrV}V��m��Uw~p�{Ywqs mns�oqpVl��#wqs�mWjy X}Votm��Uw~p"lb�9®W�zs�lnm9m�ikj(��w~pVl�mns�}V��m��Uw~p�wq{#�"ots�ot�.w~xz���5x��zu���Um�l¢�Fr"���.}"l��Uw~p�xz��u��Um�l�5{Ywqs�xz�zpkjyots�oqpVrDpVw~pJ�¤xz�zpkjyots�mns�oqpVln�.wqs�m/jy �}Votm��zw~pVl�oqpVr,m�iVjypo�u�w~u�jypfmh�bxUw~l�}ks�j�u�jbm�ikwJr��Noqlnjyr@w~p_jypkjbs�|q�_u��zpV�zu@�U�yotm��Uw~p��Vs���pV�b�U�"xUjyly�B¯hxUw~l�}ks�ju�jbm�ikwJrVl ots�j�lnm�}VrV�Ujyr�{Ywqs�u�w~u�jypfm5ln�Xl�mnjyu�l9wq{-oqpf��wqs�rVjbsy�fm�ikj�xUw��5jylnm�pkw~pk��mns��U�J�zoqx�boqlnj��Ym��%wt�¤u�w~u�jypfm��bxUw~l�}ks�j\�5xUjyoqrVl�mnw�¯�otmnm�oqpkjbw_ln�Jlnmnjyu�ly�

Q,G�°-±¢²NZ\O `qev¯�ikjyu�wqm�oc¦k�zlb�PmnsoqpVln�.wqs�mMjy X}Votm��Uw~pVlb�Bi����.jbs��.w~xz�z��ln�Jlnmnjyu�ly�³©Mjyxzxzjbsn�ªXjb|qjyxPu�wJrkjyx´�JmnjyxUjb|qsot�"i�jy X}Votm��Uw~p-�V�Nots�ot�.w~xz�z��xz�zu��zmb�Xu�w~u�jypfm/�bxzw~l�}ks�j

µ�¶W·c¸º¹�»�¼�½´¸º¾H¿¢À�ÁBÂPúÄq»«¼´¾¤Å�Æ#Ç�Å�¾�È\»�É(Å�¾�¸ºÊnÅ�Ã�Å·cË�Ìy¾¤Å�¾�¸º½F¾�¸ºÊnÅ�ÃPÌ�Ê�¸º»�·\Ê�»�½�Æ#ÍNË�É�Àηb¾�Àη�Ï-ÐÎÑ£ÒÎÑ%ÓÆÔ Å�·tÅËcÅ�Æ�Õ×ÖyØUÙ�ÙÛÚ�ÜXÝ#Þ\ßÎÙºà¤Ú�á�Õ�ß�â×ã�ßÆt½´ä\åcåqÀ¼´¾�»�Ë�Äb¿/³æWä\É(Í�Å·tË/æ9Ì�Í"ç Ô³èé�ê ¸ºÀÎÉ(Å�¾�È\»�É(Å�¿y¾�¸ÛëqÆn¶³·c¸º¹�»«¼�½´¸ì¾H¿BÀÁcÏ íä\Äc¸º·cî»�·�Æγä�Á�Ë\»«¼kÇ�À¼�îλ«·c½F¾�»�úú»WÓnïyÆÎð³ñFòÒïÎòÐPÏ�íäcÄ\¸Û·\îλ�·fÆ

ó â ô³â õ ß�ö�Ú«ÙÛÚ�áÎÝ#ÞyÜ\ØU÷�ÕUÞcÚ�à«Ø�ÜbøbÚ�Ü~â×ö�Ú�Æq½´äcå\åqÀ¼´¾�»nË/Äy¿�ÂPæWäcÉ(Í è

ù

Page 2: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

ú û ��!.�-��)�4��W!.�����

ü �X�.jbs��.w~xz�z�@u�wXrkjyx�l¢oqp"rýmnsoqpVln�.wqs�m�jy �}"otm��Uw~pVl�ots�j�}Vlnjyr��zpÿþDotm�ikjyu�otm��z�boqx/�5�U�w~xUwq|q�,mnw�u�wJrkjyxWu�w��qjyu�jypfm oqp"r,|qs�w\�(m�i�wq{��.wq�"}"xzotm��Uw~pVlb��®kwqsMj�¦koqu��"xUj���jbs�m�oq�zp�"oq��mnjbs��zo��Fxz�z¨qj������������ ����� ����������wqs��������������������� "!$#%�&�'��($��(� �9l�ikw\�­oM�qjbs�����iVots��oq��mnjbs��zlnm��z��u�w��qjyu�jypfmh�"otmnmnjbs�p�� §³jbs�UwXr"l9wq{�lnmns�oq�U|~i�m�s}VpVl oqxUmnjbspVotmnj/�/�zm�i��.jbs��UwJrVlwq{%s�oqp"rkw~u�s�wqm�otm��Uw~pVlM��iV�z��i�xUjyoqr�mnw�s�jbwqs��Ujyp�m�otm��Uw~p�wq{%m�iVj���jyx�xzl��*) +�,Y���g(iV�zl��.j��iVoy�J�Uwqs@�boqp �#j�u�wXrkjyxzjyr�����o.-������/���" "!10�(2��#3#�*�4�$���������iV�z��iÿ��pÿoS��w~pfm���p�}V}"u{Hwqs�u¢}Vxzotm��zw~p�xUjyoqrVl�mnw@o5 "6���7�8#��� ���9�(7�� "� ���;�*);:�<�,Y��gPsoqpVln�.wqs�m�u�wJrkjyxzlh�zp�w~pkjMln�"oq��j�rV��u�jypVl��Uw~p��boqpv�.jMlnjbjyp�oql(i����.jbs��.w~xz�z�/ln�Xln�

mnjyu�lb� þ�wqs�jbw��qjbsy�k�zp�oqp��¢ln�Noq��j(rV�zu�jypVl��Uw~p-�tu�w~u�jyp�m��bxUw~l�}ks�jhu�jbm�ikwXr"l9xUjyoqr�{Hs�w~umns�oqpVln�.wqs�m�u�wXrkjyx�l9mnw�i����.jbs��.w~xz���%ln�Jlnmnjyu�ly�9�«p���w~p�mns�oqlnm5mnw�r"���.}"l��Uw~p��"oqlnjyr_u�wJrJ�jyxzlb�(mns�oqp"ln�.wqs�m@u�wJrkjyxzl_oqpVr¡i��X�#jbs��#w~x��z��ln�Jlnmnjyu�lvrkwýpVwqm�l�iVw���m�ikj;}"pf�hoqpfmnjyrj��#jy��m¢wq{(�zp>="p"�UmnjyxU�,{Foqlnm��Vs�wq�"ot|~otm��Uw~p��vg³s�oqpVl��#wqs�m�jy �}"otm��Uw~pVl ots�j@�"oqlnjyrýw~p rVj��m�oq�zxUjyr��zpV{Hwqs�u@otm��Uw~p�w~p�m�}kspV�zpk|hs�otmnjq�cm�}ks�pV��pk|�rV�zlnmns��z�"}km��Uw~p�oqpVr�u�jyoqp�ln�.jbjyr-�³g/ikjs�jyxUjb�toqpfmM�"otsoqu�jbmnjbs�lM�boqp��.j�j�¦Jmns�oq��mnjyr {Hs�w~u�m�ikj�u�jyoql�}ks�jyu�jyp�m�lMwq{%��pVrV�U�J�zrV}Voqx�"otm�iVlb�gPsoqpVln�.wqs�mhu�wXrVjyxzl5{Hwqs��"�zw~xUwq|~�z�boqx.ot�V�"xz���botm��Uw~pVl5ots�j��bxUw~lnjyxU�@s�jyxzotmnjyrvmnw�mns�oqpVln�

�#wqs�m(u�wXrVjyxzl��zp�§�i��Jl��z�blb�Vxz�z¨qj�lnjyu��z��w~pVr"}V��mnwqs?) +�@A,��VsoqrV�zotm��Uw~pB);<�C�,��NoqpVr�pkjy}kmns�w~pmns�oqpVln�.wqs�m?) +�D�,��Noql��%jyxzxPoql/mnw_m�ikjbs�u�wJrk�JpVoqu��z�blE);< ù ,BoqpVr�m�iVj �5w~xUmn�yu�oqpVpDjy �}"oc�m��Uw~pF)HGI,��J);K�,����«p�m����"���boqx9�"i��Xl����boqxWot�"�"xz�z�botm��Uw~p"l�m�iVj@rV�Us�jy��m��zw~pVoqxB��iVoqpk|qjyl wq{%m�iVj�zpVrV�U�J�zrV}"oqxt�"ots�m��z�bxUjylWots�jhrks��U�qjyp��X� ��w~xzxz�zl��Uw~pVlb�Pg/ikjylnjh��w~xzxz�zl��zw~pVl�}Vl�}"oqxzxU�M��w~pVlnjbs��qju�oql�lb� u�w~u�jypfm�}Vu oqpVrýjypkjbs�|q�q��oqp"rýikjyp"��j@m�ikjv��w~xzxz�zl��Uw~p wq�.jbs�otmnwqs�iVoql�o5="�qj��rV�zu�jypVl��Uw~pVoqx�pX}Vxzx³ln�"oq��jq���«p;o@�N�Uw~xUwq|~�z�boqxW��w~pfmnj�¦Jm�w~pVxU��m�ikj¢mnwqm�oqxB�"ots�m����bxUj u�oql�l�zl(��w~pVlnjbs��qjyr-��L�m�ikjbs��/�zlnj�m�ikj r"�Us�jy��m��Uw~pVoqx-�iVoqpk|qjyl�iVoy�qj�mnw��.jMmns�jyotmnjyr,oql�ln�.w~pJ�m�oqpkjbw~}Vl��Y���Um�ikw~}km(��w~xzx��zl��Uw~pVlÎ��XoqpVrvm�iVjb�vrVw�pkwqm(pkjy��jyl�l�ots�zxU�@�"s�jylnjbs��qj�jypVjbs�|q�_wqsu�w~u�jyp�m�}Vu�� ü jyp"��j�m�ikj�m�}ks�p"�zpk|�wq�#jbsotmnwqs iVoqlMovw~pkj��¤rV��u�jypVl��Uw~p"oqxWpX}VxzxWln�"oq��jq�g(iV�zl5rV���.jbs�jypV��j��.jy��w~u�jylh�zu��.wqs�m�oqp�m5�U{-w~pVj�}Vl�jyl%{F}VpV��m��zw~pVoqxNoqp"oqxU��m����/�Vs�wq�#jbs�m��Ujylwq{Nm�ikjhwq�.jbs�otmnwqs�l��zpf�qw~xz�qjyr��Fxz�U¨qj�ln�.jy��mns�oqxX�Vs�wq�.jbs�m��Ujyly�~lnm�ot�"�zxz�zm¤�q�t�"ots�ot�.w~xz�z�5l��boqxz��pk|xz�zu��Um�ly�Joql��Xu��"mnwqm��z���.jyiVoy�J�Uwqs(jbm��t� ��¥ x�xWu�w��J�zpk|�ln�.jy�b�Ujyl�wqs��Ujyp�m�m�ikjy�UsMu�w��qjyu�jypfm�w~p�j�¦Jmnjbs�pVoqx9��pk{Hwqsu�otm��Uw~p-�M®kwqs

�"oq��mnjbs��zoDm�ikj�w~pVxU�Slnw~}ks��j_wq{/��pk{Hwqsu�otm��Uw~p ��w~u�jyl��J�zo,m�ikjy�Us���jyxzx%l�}ks�{Foq��j_s�jy��jb�V�mnwqs�l��/iV�z�i,��w~xzxzjy��m���pk{Hwqsu�otm��Uw~p,u@oq�zpVxU��w~p;��ikjyu@�z�boqxB�b}kjylb���«p��ikjyu�wqm�oc¦k�zlb�.{Hwqsj�¦Joqu��NxUjq�9��jyxzxzl¢u�w��qj�mnw��hots�rVl�iV�U|~iý��w~pV��jyp�mns�otm��Uw~pVl¢wq{/oD��ikjyu����boqx%otmnmns�oq��m�oqp�mb�M ¦J�#jbs�zu�jyp�m�l�w~p ��iVjyu�wqm�oc¦J��l u�jyoql�}Vs�j�jy�Um�ikjbs m�ikj��.jyiVoy�J�Uwqs�wq{hm�ikj��.wq�"}"xzotm��Uw~poql�oD��ikw~xUjq�9jq�ì|k� �zpýmnjbs�u�l�wq{�rkjypVl��Um��zjyl��Yjq�ì|k� ¬ w�wJrk�hots�rN�� E���POQ);:RG�,Y� wqs¢m�iVj�"otm�iVl�wq{%�zp"rV�U�J�zrV}Voqxzlhots�j�{Yw~xzxUw\�%jyr;�/�Um�i;o��X��rkjbw�ot�"�"ots�otm�}Vl��F®kwqs�rF�4 S���POT) ù <A,Y��L�{%��w~}kslnj�m�ikj��.jyiVoy�J�UwqsMwq{5m�ikj��.wq�"}Vxzotm��zw~p,s�jyl�}VxUm�l�{Ys�w~u m�ikj�u�w\�qjyu�jypfm�wq{5�zm�lu�jyu �.jbs�ly�,þ,otm�ikjyu@otm��z�boqxhu�wXrVjyxz�zpk|��"s�w��J�zrkjyl�o��hoy�Smnw;s�jyxzotmnj�m�ikj_��pVrV�U�J�zrV}VoqxoqpVr���w~xzxUjy��m��z�qj u�w��qjyu�jypfm��"ots�oqu�jbmnjbs�ly�Vjq�ì|k���X�_{Ywqs�u���pk|��"ots�ot�.w~xz�z��xz�zu@�Um�lb�

D

Page 3: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

g(ikj�=Vs�l�m�u@otm�ikjyu�otm��z�boqx u�wXrkjyxW{Ywqs ��iVjyu�wqm�oc¦J��l��zl�m�ikj1US�R "���AVAW X����'�Y���WZ���8[2���u�wXrVjyx.�F§%©�ªV��t�/iV����i��zlW�"oqlnjyr¢w~p��5s�w���pV�zoqp�u�wqm��Uw~p��³�«m9�zlW¨Xpkw\�/p¢m�iVotm9m�ikj�§�©¢ªu�wXrVjyxXl�ikw��/lBoqxzxX¨J�zpVrVl³wq{Nr"���#jbs�jypfmW�"otmnmnjbs�pVly�qjq�ì|k� lnm�oqpVrV�zpV|/�hoy�qjylb�Xot|q|qs�jb|~otm��Uw~p-�="pV�UmnjMm��zu�jM�"xUw��(�¤}k���Jwqs/ln�"��pkwXr"oqx#rVjy��w~u��.w~l��Um��Uw~pv�"otmnmnjbspVl �*) +�\�,Y���«pDm�iV�zl�ots�m��z�bxzj �5j�s�jb�J�Ujb� if�X�.jbs��.w~xz�z�Mu�wJrkjyxzl�oqpVrDmns�oqpVln�.wqs�m�jy �}Votm��zw~pVl/mnwt�

|qjbm�ikjbs �/�Um�i<m�ikjy�Us�ot�V�"xz�z�botm��zw~p mnw£��ikjyu�wqm�oc¦k�zlb� ¬ jÿ��w~u��"ots�j3m�ikj3s�jyl�}Vxzm��zpk|��ikjyu�wqm�oc¦k�zl;u�wJrkjyxzlvmnw­m�ikjSl�m�oqpVrVots�r §%©�ª���m����.jSu�wJrkjyxzl�oqpVr �%jSl�iVw�� m�i"otm�zp���jbs�m�oq�zp l��Um�}"otm��Uw~pVl�m�ikjb�Di"oy�qj�oqrk�toqpfm�ot|qjyl �/iV�zxzj��zp;wqm�ikjbs�l�m�iVjb�Dots�j�jy X}VoqxzxU�ot�V�"xz�z�bot�NxUjq�;�5jb{Ywqs�j�rkjyl���s�U�"�zpk|Dpkw~pJ�¤xz�zpVjyots¢�ikjyu�wqm�oc¦k�zl�u�wJrkjyxzl��%j]=Vs�l�m�s�jy�boqx�xl��zu��"xzjbshx��zpkjyots(�boqlnjylb�

^`_a^ bTcedRd�f%g8h�i�f%jkdRh`lmjncpo q]hrgasutalvcwlmfx�jJtaoyf%lJzAtacwlþ�w\�qjyu�jypfm(��p�w~pkj�l��"oq��j�rV��u�jypVl��Uw~p��/�Um�i���w~pVl�m�oqpfmhln�.jbjyr�{�oqpVr_��w~pVlnm�oqp�m%m�}kspJ��zpk|�s�otmnjT|¡�boqp �#j_rkjyl���s��U�.jyr �X� o,��wqs�s�jyxzotmnjyr soqpVrkw~u&�hoqxU¨#��g(ikj�mnwqm�oqx%�.wq�k�}Vxzotm��Uw~pýrkjypVl��Um��~}W�����*�.� �zl l��"xz�Um oql?}F��}r���Q}w�R�zp�mnwDrVjypVl��Um��Ujyl�{Hwqs¢s��U|~i�m$�cxUjb{Ymu�w��J�zpk|��"ots�m5wq{�m�ikj��.wq�"}Vx�otm��Uw~p-��}r�J�*}e�9�Js�jyln�.jy��m��U�qjyxU�q� g(iVjylnj�ots�j�m�iVj��tots��zot�NxUjylwq{Wm�iVj?��w~xzrVlnmnjy��pJ�¤©�oq��u�wJrkjyx.{Ywqs/��wqs�s�jyx�otmnjyr�s�oqpVrVw~u��5oqxU¨ �*) ù @�,���) +�+�,Y��} �� �T{�} �� ����|`} � � | D ��} � ��} � �$� } �� ��{�} �� ����|`} � � | D ��} � ��} � �$��� ù �g(iV�zl�ln�Xl�mnjyu��boqp;�.j mnsoqpVln{Ywqs�u�jyrDmnwvoqp;jy X}V�U�toqxUjyp�m�ln�Jlnmnjyu�{Hwqs�m�ikj¢mnwqm�oqxB�.wq�k�}Vxzotm��Uw~p�rkjyp"l��Um¤��}Doqp"r�m�ikj��.wq�"}Vx�otm��Uw~p��"}J¦T���Q{9��} � �~} � �$�

} � �3� � ��C� � � ��{ [ } � ����|`��� � D~��%��jyx��zu��zpVotm���pk|�m�iVj��cots�zot�"xUj�� �8X��R�%�7 "$� ��Vy�hw~pkj�wq�"m�oq�zpVl(o@mnjyxUjb|qs�ot�"i�jy X}Votm��Uw~pwqs�rVoqu��.jyrv�hoy�qj�jy �}Votm��zw~p-�

ù| } �P� �3} � � { [| } ��� � � K~�� wqmnj(m�i"otm9m�ikjhmns�oqpVl��Um��zw~p�{Ys�w~u�� ù �³mnw@� K~���Y�/i"�z��i��boqp��.jh|qjypkjbs�oqxz�U�bjyr¢mnw�ln�Jlnmnjyu�l�/�Um�i,l�jb�qjbs�oqx³rVjb�#jyp"rkjypfm��tots��zot�NxUjyl(�zpDoqpf��ln�Noq��j�rV�zu�jypVl��Uw~pN�%�zl�pkwqm���w~u��"xUjbmnjyxU��zpf�qjbs�m��U�"xUj��*);DA+R,Y����{��5jvxUjbm�m�iVj@�"ots�oqu�jbmnjbs�l�|¡oqpVr�{­|qwDmnw��zp>=NpV�Um¤��l�}V�i m�i"otm�m�ikjv X}kwqm��Ujyp�m{ [/� |­��w~p��qjbs�|qjyl¢mnw,o�pX}Vu �.jbsE� �¡C�m�ikjyp �5j�{Hwqsu�oqxzxU��wq�"m�oq�zp m�ikj��Nots�ot�.w~xz�z�

xz�zu��Um�} � �u�T} ��� � � wqm�����j�m�iVotm��zp�m�i"�zl/xz��pkjyots/�boqlnj¢m�ikj mnsoqpVl��Um��Uw~pD�zl(jy X}V�U�toqxUjyp�mmnw�o�l��boqx��zpk|/wq{Nln�"oq��j�oqpVr¢m��zu�j�¢T��£ [ �Poqp"r?¤¥��£/���/�Um�i¥|�oqp"r¥{@iVjyxzr���w~p"lnm�oqpfmb�

K

Page 4: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

^`_'¦ §T¨mf3©�talmf7hrd�§ªdRh`lJzI«�cwd�i5¬�­¯®nh�iRt8cwl¬ j�rVjypkwqmnjDm�iVj,�.wq�"}Vxzotm��zw~p­rkjypVl��zm¤�­otm�m��zu�j5��°±CJ�(�.w~l��zm��Uw~p.�³²��H´�µ"�/oqp"r�qjyxUwX�b�zm¤�3�Q²k¶¸·��H´SµS���Um�iº¹W���$�*�p�*�J�� ¬ j,oql�l�}"u�jvm�iVotm�m�ikj�lnjbm�wq{��.w~l�l��U�"xUj�qjyxUwX�b�zm��Ujyl�¶³·ÿ�H´ µ �zl5�.w~}VpVrVjyr_oqpVr�l��Xu�u�jbmns��z���F�´�ìjq�»�]²º¶k¼½����²5¶��� g(iVjypm�ikj�xz�zpkjyots�mns�oqpVl��#wqs�m�u�wXrkjyx��#��iV�z��i��zl��"oql�jyr�w~p o_�qjyxzwX�b�Um���¾�}Vu����Vs�wX��jyl�l��Flnjbjjq�ì|k�hªXmns�wXwJ��¨B);:�<A,�wqs¿L�m�iVu�jbs?�� À���PO¿);<�KA,Y�5s�jyoqrVlÁÁ � ¹W�����*�w�*�k�w����Â/ê¹W�����*�w�*�k�m����|¹B���$�*�p�*�J�e�3|?ÄÆÅ¢�����*��Ç��ȹW�����*�w�*��ÇH�ZÉ���Ç"� ��+��

ü jbs�jT| �zlMm�iVj�m�}ks�pV��pk|vs�otmnj@wqs�m�}Vs�pV�zpk|�{Ys�jy X}kjypV���q�WoqpVrB¢�� ù/� |¡��lMm�ikj@u�jyoqps�}Vp�m��zu�jq�hg/ikj�¨qjbs�pkjyxeÅ¢���%�*� Ç ��rkjyl���s�U�.jyl�m�ikj��Vs�wq�"ot�"�zxz�zm¤�@{Ywqs/m�iVj�pkjb�1�qjyxUwX�b�zm¤��_|~�U�qjyp�m�ikj��Vs�jb�J�Uw~}Vlh�qjyxzwX�b�Um��]� Ç �"ikjypV��j¥ÊªÅ����%�*� Ç �ZÉ���� ù �g(ikj�w~pkj/rV�zu�jypVl��Uw~pVoqx%��w~xzrVlnmnjy�zpk�¤©�oq�(u�wJrkjyx³� ù � wJ�b�b}ks�l�oql5o l��#jy�b��oqxV�boqlnj��zp

w~pkj ln�"oq��j�rV�zu�jypVl��Uw~pv�U{P�5j lnm�}VrV�;��+��h���Um�i]��²ÌËAͪ{wÎ�oqpVrÏÅ¢���%�*� Ç �»Ð ù/� DX�^`_'Ñ ÒÓc»jnf%gÔz]qºt8i�¨kÕ�f7h`Ö2iRt8cwl´(jyoq��m��Uw~p�r"���.}"l��Uw~p�jy X}Votm��Uw~p"l9ots�jhm�ikj�lnm�oqpVrVots�r�u�wJrkjyxzlW{Ywqs ln�Vs�jyoqr��zp¢ln�"oq��j(oqp"r�zpfmnjbsoq��m��Uw~pÿwq{��"ots�m��z�bxzjylb��jq�ì|k� rVjypVl��Um��f�¤rkjb�.jypVrVjypfm¢�"�Us�m�iÿoqp"rÿrkjyotm�iÿ�Vs�wJ��jyl�lnjyl�Yjq�ì|k�,þ,}ks�s�oy�F);<�DA,Y��v�«{hm�ikj@rV���-}Vl��zw~p��Vs�wJ��jyl�l �zl�s�jb�"xzoq��jyr ��� o,u�wqs�j@rkjbm�oq�zxzjyrmns�oqpVln�.wqs�m��Vs�wX��jyl�ly�km�ikjyp�s�jyoq��m��Uw~p,mns�oqpVln�.wqs�m�u�wJrkjyxzlhs�jyl�}VxUmb�

¹ � �3�¥ÂIÿ¹]����|¹��3|?ÄÆÅ¢�����*� Ç �ȹW�����*�w�*� Ç �ZÉ�� Ç �Ø×T) ¹7,8� � <~��/ikjbs�jMm�ikj {Y}Vp"��m��Uw~pVoqx`×1) ¹7,PrVjyl���s��U�.jylhm�ikjM|qs�w\�(m�i,rV�XpVoqu@�z�bl��Flnjbj�);D�K�,Y��¬ iVjbs�jyoql��zp;jyots�xz�zjbs��Not�#jbsl�w~p�s�jyoq��m��Uw~p�mns�oqpVln�.wqs�m�jy X}Votm��Uw~pVl��FoqpVr�w~p ��wqs��

s�jyxzotmnjyr s�oqp"rkw~u �hoqxU¨;u�wXrkjyx�l��/�Um�i;s�jyoq��m��Uw~pVlÎ��m�ikj@pkw~p"xz�zpkjyots��zm¤�,iVoql pkwqmM�.jbjyp{Y}ks�m�ikjbs�ln�.jy�b�P="jyr-�Nxzotmnjbs��N�/�zm�i�s�jyln�.jy��m�mnw_jy��w~xzwq|~�z�boqxBu�wXrVjyxz�zpk|k�#o@�bxUjyotsMrV�zl�m��zpV�Î�m��Uw~p��.jbm¤�5jbjyp��Vs�wXrV}"��m��Uw~p�oqpVr�s�jyu�w\�coqxVjb�qjypfm�l5iVoql9�.jbjyp�u�oqrkjq�B�«{��f{Hwqs j�¦koqu��"xzjq�m�ikj�pkjb�/xz�/�Vs�wJrV}V��jyrM��pVrV�U�J�zrV}Voqx�l"iVo��qj%o(}Vp"�U{HwqsuRr"�zlnmns��U�N}km��Uw~p�wq{X�qjyxzwX�b�Um��zjylb�\m�iVjypm�ikj pkw~pVx��zpkjyots��Um��@iVoql�m�iVj�{Hwqsu

×1) ¹7,�� ùÙ ¶ Ù"Ú ��ÛÝÜ��ÔÛÝÜÀ��Þ#��ÛÝÜ\�ȹp��/�Um�i pVw~pJ�¤pkjb|~otm��U�qj�s�otmnjyl Ú ��Û Ü �$�6Þ#��Û Ü ��Dg/ikj�mnwqm�oqx5�.wq�"}Vxzotm��Uw~p rkjypVl��Um��S�zl rVj��pkwqmnjyr,oql ÛÝÜ~���$�*�-�n�NÄ�ß¿¹W���$�*�p�*�J�2É�� � :~��Y�%j��/�zxzx.}Vlnj�Û1à�{Hwqs�u�w~u�jyp�m�l/xzotmnjbs���%g(iV��l�u�wXrVjyx-�Vs�jylnjbs��qjyl(�.w~l��Um��U�J�Um��q�9�«p�m�iVj�zlnwqmns�wq�"�����boqlnjq�³�/ikjbs�j¢m�ikj�pkw~pk�¤xz�zpkjyots��zm¤��rkjb�.jypVrVl�w~pVxU�,w~p�Û Ü �e×1) ¹7,»�âá9��Û Ü ��

+

Page 5: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

�#w~l��Um��U�J�Um¤���boqp�pkwvxUw~pk|qjbs��#j�|~}Vots�oqp�mnjbjyr �U{náS�iVoqpk|qjylMl��U|~p���®kwqsM�zlnwqmns�wq�"�z� oqp"rpkw~pJ�¤�zlnwqmns�wq�"�z�(u�wJrkjyxzl��.w~}VpVrVots����coqx�}kj�oqpVr_ln�.jy��mns�oqxV�"s�wq�"xUjyu�l%iVoy�qj��.jbjyp@lnm�}"rJ��Ujyr��zp~);DA+R,�� g/ikjMmns�oy�qjyx��zpk|�{Ys�w~pfm/�Vs�wq�"xUjyu@l5��pvm�ikj��boqlnj�wq{Bw~pkj�ln�Noq��j rV�zu�jyp"l��Uw~p�Y�/iV�z�i���w\�qjbs�lBoqxzlnw�m�ikj%�boqlnj5wq{#¯hotmnm�oqpVjbw�l��Xlnmnjyu@l³�zp�oqpf��l��"oq��j5r"�zu�jypVl��zw~pN�#iVoy�qj�#jbjyp�lnm�}"rV�Ujyr¢�zp�o�lnjy X}kjypV��jhwq{V�Not�#jbslb�t�/ikjbs�j�);D�<A,k|~�U�qjylWm�ikj�u�w~lnm9rkjbm�oq�zxzjyr�oqpVoqxU��Xl���lb��®VwqsMm�ikj¢l�m�ot�"�zxz�Um���wq{�i����.jbs��.w~xz�z��{Ys�w~p�m�l�lnjbj�oqxzlnw~) ù \A,�� ªJ��i��%jbm�x��z��¨Ø);: ù �r:�D�,iVoqlhwq�Vm�oq�zpkjyr�s�otm�iVjbs�|qjypkjbs�oqx.s�jyl�}"xUm�l5w~p_m�iVj�u@�zpV�zu�oqxN�Vs�wq�Not|~otm��Uw~pvln�.jbjyr���p_s�j��oq��m��Uw~p�mns�oqp"ln�.wqs�m�jy �}"otm��Uw~pVl �Y�/i"�z��i�rVjb�#jyp"rVl5w~p,ln�"oq��j�rV�zu�jypVl��Uw~pv�zp�o�pVotm�}ksoqx�5o��k�(oqpVr�w~p�m�ikj j�¦J�zl�mnjypV��jMwq{B{Hs�w~p�m�lb�

^`_Yã Õ�f%zIi�tÔlnäyz/i�h�i�f%z�«p�s�jyoq��m��zw~p­rV���-}Vl��Uw~pýjy X}Votm��Uw~pVl�oqp"r�s�jyxzotmnjyrÿlnjyu���xz�zpkjyots�mns�oqpVl��#wqs�m¢jy X}Votm��Uw~p"lm�ikj mns�oqp"ln�.wqs�m/oqpVr,m�ikj��zp�mnjbs�oq��m��Uw~pD�Vs�wJ��jyl�lnjyl/s�}Vp,�"ots�oqxzxUjyx�� ü w��5jb�qjbsy���zp�u�oqp��s�jyoqxz�zlnm�����l��zm�}Votm��Uw~pVl�m�ikjb�;oqxUmnjbs�pVotmnjq�-jy�Um�iVjbsM�#jbs�UwXr"�z�boqxzxU��wqsM�/�Um�iDs�oqpVrkw~u�mns�oqpk�l��Um��Uw~pVly�v§Wjbs��zwXrV���boqxzxU��oqxUmnjbs�pVotm��zpV|,oq��m��Uw~pVl¢xUjyoqrSmnw�pkw~pJ�¤oq}Vmnw~pkw~u�w~}Vl��Vs�wq�"xzjyu�l�*);D�@�,Y���/iV��xUj�s�oqpVrkw~u mns�oqp"l��Um��Uw~pVl�xUjyoqrSmnw,o�pkjb���bxzoql�lMwq{%s�jyoq��m��Uw~p rV�U�.}Vl��Uw~p,wqsmns�oqpVln�.wqs�m¢ln�Xl�mnjyu�lb�Dg%�X�"�z�boqx%j�¦Joqu��NxUjyl�ots�j�ln�Jlnmnjyu�l¢rkjbs��z�qjyr {Ys�w~u m�ikj_®B�zl�iVjbsjy �}Votm��zw~pý�*) +&GI,��p);D�\�,Y�

� � � ��å��?��{ [ �E��{ K6ææ � � á9� æ ����{ Kçæ ��{ [ � �8Gq��/�Um�iÏá9� æ �n� æ � ù � æ ��Vrkjyl���s�U�"�zpk|¢o�u�w��J�zpk|�lnm�otmnj���oqpVr�oqp��zp�mnjbs�oq��m��zpk|�lnm�otmnjæ � g(iV�zl�l��Xlnmnjyu7�zl@jyl�lnjyp�m��zoqxzxU�ÿjy X}V�U�toqxUjyp�m�mnwÿoS{Ywqs�u�oqx��5o��qj�jy �}Votm��zw~p¡���Um�ipkw~pVxz�zpVjyots�rV���-}Vl��Uw~p�oqpVr��/�zm�iÿ�X��l���w~}Vl�rVoqu��N�zpk|­�Y�/�Um�iè¢�� ù/� �'{ K �Æ{ [ ��oqp"ré�ê ��¢2{ ê � ¢ æ �P� � � ù �B¢7á�ÇY� æ �n� æ � �B¢>�]å æ �� é K ��å æ � é [ á9� æ �¯�~¢>�]å�á9� æ �$� � \~��«pvm�iV��lhjy �}Votm��zw~p�ot|~oq�zp�o~ë��"ots�ot�.w~xz����xz��u��Umçì�¢1íîC��boqpv�.j�m�ot¨qjyp��/iV����i�s�jyl�}Vxzm�l�zp�oqpkwqm�ikjbs�s�jyl��boqxUjyr�®B�zl�ikjbs�jy �}"otm��Uw~p-�g(ikj �hoy�qj%wq�.jbs�otmnwqs³�zp�m�i"�zl-�Vs�wq�NxUjyu����´�ìjq�U� æ �P� � é K ��å æ �E¢>��å�á9� æ ��\l�iVw��/l³omns�oqpVl��Um��zw~pM{Hs�w~u if�X�.jbs��.w~xz�z�³mnw(jyxzxz�z�Vm��z�W{Hwqs³xzots�|qjm¢.�Bþ�wJrkjyxzl-wq{Jm�iV��l-{Hwqsu oqxzlnw(�Nxzoy�

o@s�w~xzj¢�zp,m�ikj¢m�ikjbwqs���wq{���pk{Hjy��m��zw~}Vl�r"�zlnjyoqlnjylb�#�/ikjbs�j m�ikj�u�w��J�zpk|voqpVr;lnjyrkjyp�m�ots��lnm�otmnjyl���wqs�s�jyln�.w~pVr�mnwvxUw~pk|@s�oqpk|qj�oqp"r,l�ikwqs�m(s�oqpk|qj¢�zpk{Yjy��m��Uw~pVl?);D�:�,��hg/ikjylnj �Nikj��pkw~u�jypVo�oqpVr���w~pVpkjy��m��zw~pVlPl�}V|q|qjylnmPmnw�lnm�}Vrk��mns�oqpVl��#wqs�m-jy �}Votm��zw~pVl��/�zm�i�lnjyrkjyp�m�ots��lnm�otmnjyl��zpv|qjypkjbs�oqx�);K�KA,��5g/ikj ln�.jy��mns�oqx-m�iVjbwqs���oql/rkjb�qjyxUwq�.jyrD�zpDªXjy��m��Uw~pºK��bots�s��zjylw��qjbs�mnwvmns�oqpVl��#wqs�m�jy X}Votm��Uw~pVl��/�zm�i,s�jyl�m��zpk|��"iVoqlnjyl�oqpVr;rV���-}Vl��Uw~pDxz�zu��zm�l/�boqp��.jlnm�}VrV�Ujyr@�zp�m�iVotm�{Ys�oqu�jb�5wqs�¨Ý);K�KA,��B�«m m�}ks�pVl w~}km m�iVotm�o�ln�"otm��zoqx�xU��rkjb�.jypVrVjypfm lnmnwq�V��"�zpk|�s�otmnjE{ [ ���.�(xUjyoqrVlb�V�zpvm�iVj��"ots�ot�.w~xz�z��xz��u��Umb�Jmnw@o�rks�U{Hmhmnjbs�u �Vs�wq�.wqs�m��Uw~p"oqx.mnw

<

Page 6: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

Ã�{ [ ���-�� g(iV�zl rVjyl���s��U�.jyl o¢pkjbm%rVs��U{Ym9mnw��hots�rVl5{Yo��qwqs�ot�"xUj�iVot�"�zm�otm�l��Ys�����i@pX}kmns��zm��Uw~pwqs�l�ikjyxzmnjbsÎ���«pD|qjypkjbs�oqxB�Um��boqp;�.j�lnm�otmnjyr;m�iVotmM�zp�mns�wJrV}V�b�zpk|�o_s�jylnm���pk|_lnm�otmnj��zp�mnwvo@|~�U�qjyp

rk�Xp"oqu��z�bl¢mnjypVrVl�mnw s�jyu�w��qj�w~l��b��xzxzotm��Uw~pVly�  �}V�zmnjv��p3��w~p�mns�oqlnm�mnwýo;rVjyxzoy�3�/iV�z�i�Ymnwq|qjbm�ikjbs(�/�Um�i�pkjb|~otm��U�qj�{Yjbjyrk�"oq��¨k�%mnjypVr"l�mnw�jypViVoqp"��j�w~l��b��xzxzotmnwqs����.jyiVo��X�zwqsy�Xl�jbjo�{Hwqs�m�iV��w~u��zpk|��Not�#jbs¥);DRGA,��ï 6 ��)�����vðy���òñ£8��$ ��!#��ó������{(m�iVjvu�w��qjyu�jyp�m�wq{�o;�.wq�"}Vxzotm��zw~pSwqs�oqpÿ�zpVrV�z�X�zr"}VoqxB�zl¢�"�zoqlnjyrý���3oD�ikjyu��z�boqxl��U|~pVoqx-m�iVj�s�jyln�.w~pVlnj���l�mnjbs�u�jyr3��������R 8�4ô��'� �Ywqs�u�wqs�j |qjypkjbs�oqxzxU�B��������A�4���>���� "��-������-����1���% ���Dþ,wXrkjyx�lM{Hwqs���iVjyu�wqm�oc¦J��l�iVo��qj_�.jbjypýl�}V�b��jyl�ln{F}Vxzxz�Dot�V�"x��Ujyr�mnwD�Noq�Î�mnjbs��zoJ�Pl�x��zu�j�u�w~xzrVly�.l�¨X�zp��N�U|~u�jyp�m�otm��Uw~p��"otmnmnjbs�pVly�PxUjy}k¨qwJ����mnjyl oqpVr u�oqp��Dwqm�ikjbsj�¦Joqu��NxUjylb�¥ l u�jypfm��Uw~pVjyr'��p<m�ikj��zp�mns�wJrV}V��m��Uw~p1m�ikj3=Vs�lnm�u�wJrkjyx {Ywqs ��ikjyu�w~lnjypVl��Um��z�qj

u�w��qjyu�jyp�mDi"oql��.jbjyp rkjb�qjyxUwq�.jyr£���£§Botm�xzot¨õ);<RGI, oqpVr ©Mjyx�xUjbs�oqpVr<ªXjb|qjyx1) +�<�,��§Wotm�x�ot¨�öºl�u�wJrkjyx-�zlh�"oqlnjyr�w~p�o�rkjbm�oq�zxzjyr�s�oqpVrkw~u��hoqxU¨vrkjyl���s��z�Vm��Uw~pB);<RGI,��g(ikj�©MjyxzxUjbsn�«ªJjb|qjyx�u�wJrkjyx���pS�Um�l�|qjypkjbs�oqx%{Hwqs�u ��w~pVl��zl�m�l wq{�{Yw~}ks���w~}V�"xUjyrSs�j��

oq��m��Uw~pDr"���.}"l��Uw~p�jy �}"otm��Uw~pVlb�%�«p�u�w~lnm��boqlnjyl��Um/�zl(s�jyrV}V��jyr�mnw_m¤�5w@jyl�l�jypfm��zoqxP�cots���ot�"xUjylb��m�ikjD�.wq�"}Vxzotm��Uw~p rkjypVl��Um��Ø}W�����*�.�@oqpVr­m�ikj���w~pV��jyp�mns�otm��Uw~pRwq{�o �ikjyu��z�boqxl��U|~pVoqxr�.���$�*�-��%g(ikj�©�jyxzxUjbsn�«ªXjb|qjyx-u�wXrkjyx�s�jyoqrVl

} � � Ã�� ÷ K ��}w�*�k�6Ã�}���÷ [ ��}w�*�k�6Ã��J�� � � ÷�øçå��?��÷�ù~���J�Ô�¥�3}rá9���J�$� � @~�g(iV�zlhln�Jlnmnjyu�iVoql��.jbjypvlnm�}Vr"�Ujyr�w~p�}Vpf�.w~}Vp"rkjyr_oqp"rvw~pv�#w~}"pVrkjyr�rkw~u�oq�zpVlh���Um�i�cots��zw~}Vl¢�#w~}"pVrVots��S��w~pVr"�Um��Uw~pVlv�"ú��Us��z�iVxUjbmb� � jy}Vu�oqpVp���u���¦JjyrN�� ��pým��%wýl�}ks��qjb�ots�m��z�bxUjylb� ü wqs�l�m�u�oqpVpÏ);K�@J��+RC�,k|~�U�qjyl oqp�oqu��"xUj5s�jb�X�zjb�ÿwq{"m�ikj�u�oqp��¢pVw��­oy�toq�zxzot�NxUjoqpVoqxU�Xm��z�boqx(s�jyl�}VxUm�lb�h�zpÿ�"ots�m��z�b}Vxzots�w~p �"xUw��(�¤}k� �zp.=NpV�Umnj�m��zu�jq�R��pÿm�ikjD�boql�j,wq{��w~pVlnm�oqp�mB��wXjb�@�b�Ujyp�mW{Y}Vp"��m��Uw~pVlp÷ K ��÷ [ ��÷ ù �toqp"r�á@�Um³�zlP¨XpVw��/p�m�iVotmWm�ikj5 X}Voqxz�zm�otm��U�qj�#jyi"oy�J�Uwqs�lnmns�w~pk|~xz�Drkjb�.jypVrVl�w~p�m�ikj�l��"oq��j�rV�zu�jyp"l��Uw~p-����p ù �aú�m�ikj�ln�Jlnmnjyu&i"oql|~xUwq�"oqxzxU�_j�¦J��lnm��zpk|�lnw~x�}km��Uw~pVlb� g/ikj¥D\�aú��boqlnj ��p�oqu �"�z|~}kw~}Vl(oqpVr�m�iks�jyl�ikw~x�rVl�iVoy�qj�#jbjypv{Yw~}VpVr��W�«{Pm�iVjMmnwqm�oqx��zpV�Um���oqx"u�oql�l(j�¦J��jbjyrVl/�Um�lhm�iks�jyl�iVw~xzr-�Xm�ikjypvm�ikj�lnw~xz}Vm��Uw~p�"xUw���l/}k�;�zpÏ="pV�zmnjMm��zu�jq�h��{Bm�ikj¢�zpV�Um���oqx-u�oql�l/��l(�.jyxUw��1m�iV�zl�m�iVs�jyl�ikw~xzr��Vm�ikjyp�m�iVjlnw~xz}km��Uw~pvj�¦k�zlnm�l(|~xUwq�"oqxzxU���zpvm��zu�jq�g(ikj��"xzw����¤}k�­l�w~xz}km��Uw~pVl�wq{�m�ikj�l��Xlnmnjyu � @~��l�iVw���m�ikj,j�¦J�zl�mnjypV��j�wq{Mo �qjbs��

lnmns�w~pk| ��pVlnm�ot�"�zx��Um¤� oqp"r�o�xzots�|qj(ot|q|qs�jb|~otm��zw~pVoqx"{Ywqs���jq�9��p���jbs�m�oq�zp_l��Um�}Votm��Uw~p"lb�~ikw��(�jb�qjbsy�k�Um��zl%rkjyl��Us�ot�NxUjhmnw�wq�Vm�oq��p@lnm�ot�"xzj/ot|q|qs�jb|~otm��Uw~pv�"otmnmnjbspVlb�f��iV�z��i_rkw pkwqm%�"xUw\�}k� �zpØ="p"�Umnjvm��zu�jq� g(iVjbs�j�ots�j��tots��Uw~}Vl�u�jy��iVoqpV��l�u�l��/i"�z��i��Vs�jb�qjypfm��NxUw�� }V���g(ikjylnj��boqp,�#j��bx�oql�l��P=Vjyr�oql({Hw~xzxzw��/l��

:

Page 7: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

û ? ���� "(6�� "� ���Æ�"üª���4 �� �zp3÷ [ ��}w�*�k��wJ�b�b}ks¢�qjbs�� pVotm�}Vs�oqxzxU� �U{���jyxzx5l�}ks�{Foq��j@s�j����jb�Vmnwqs�¨X��pkjbm��z�bl ��l m�ot¨qjypÿ�zp�mnwDoq�b��w~}"pfmb�ý¯hiVjyu�wqm�oc¦J��l�u�wJrkjyxzl �/�Um�i3l�otm�}ks�otm��zw~pj��#jy��m�l_iVoy�qj��.jbjypÿlnm�}"rV�Ujyr�oqpVoqxU�Xm��z�boqxzxU�ýoqpVrÿiVoy�qj,�#jbjypÿ}"lnjyrÿ�zp�u�oqp��3ot�V�Nxz����botm��Uw~pVlv�8L�m�iVu�jbs¢oqpVr3ªXmnjb�qjyp"l�);<�<�,��W�h�zxUjbs]);<I,��»´/�U�qjbs�w.�� ¿���PO�);<�@A,��9®kwqsry�� ����PO) ù GI,Y��ý ? ¥ -����þ(�1��ÿn�����'�&[�j��#jy��m��hoql/�zp�mns�wJrV}V��jyrv�X� ü �zxzxzjyp�oqp"r,§Boq�zp�mnjbs�);K�<J��<�:�,��ü jbs�j¢�Um(�zl(oql�l�}Vu�jyr�m�i"otm/�"ots�m����bxUjyl(iVo��qj�o�="p"�UmnjM�qw~xz}Vu�j�oqpVr�m�iVotm���jyxzxzl(�boqpVpVwqmu�w��qj@��pfmnw�s�jb|~�zw~pVl��/iV�z�i ots�j�oqxUs�jyoqrV�5="xzxzjyr,�X�Dwqm�iVjbs ��jyxzxzlb� ¥ l��zu��"xUj¢�qjbs�l��zw~pwq{5m�iVj��qw~xz}Vu�j1="xzxz�zpV|�u�jbm�iVwXrSxzjyoqrVlMmnwDo�mnjbs�u ÷ [ ��}w�*�J��� � }W� ù �Ø}-��@�«m��5oqll�ikw���pSoqpVoqxU�Xm��z�boqxzxU�;m�iVotm�m�i"�zlM{Ywqs�u wq{S÷ [ xUjyoqr"lMmnw,|~xUwq�NoqxzxU�,j�¦k�zlnm���pk|�lnw~xz}km��zw~pVl�zp�oqxzx�ln�Noq��j rV�zu�jyp"l��Uw~pVlb�� ?�� (7���(�Ó�����>�$���&[MwJ�b�b}ks�l/�U{Pm�ikj���jyx�xzlhs�jyxUjyoqlnj�oqpvj�¦Jmns�o��ikjyu��z�boqx��/iV�z�i���ls�jb�"}Vxzl��U�qj/mnw�wqm�ikjbs(��jyx�xzl�);K�<A,���g(ikj�s�jyl�}"xUm��zpk|�jy �}Votm��zw~p�iVoql5m¤�5w���w~u��.jbm��zpk|�rks��U{Ymmnjbs�u�lb���ikjyu�wqm�oq��m��z��otmnmns�oq��m��zw~pSoqpVr; �}kwqs}Vu lnjypVl��zpV|@s�jb�"}"xzl��Uw~p-����m��zl�oqp;wq�.jypu�otm�ikjyu�otm��z�boqxh�"s�wq�"xUjyu mnwÌ=NpVrS|qjypkjbsoqx5��w~p"rV�Um��Uw~pVl�l�}V�iým�i"otm�l�w~xz}km��Uw~pVl �"xUw\�}k���kwqs(j�¦k�zlnm�|~xUwq�NoqxzxU�q�� ? ¥ xzl�w oÝÿn�%�� a�Ý������#��þ���&[Ø*������(2�@xUjyoqrVl�mnwý|~xUwq�"oqx(j�¦k�zlnmnjyp"��jq�hotm�xUjyoqlnm��zpD\�aú��9oql¢�hoql�l�ikw���p �X� ü ��xzxUjyp oqp"r�ªJ�iVu�jy�zl�jbs]);KRGI,�� ü jbs�j��Um��zl¢oql�l�}"u�jyr m�i"otm�zpVrV�U�J�zrV}"oqxzl�u�jyoql�}ks�jDm�ikj,��iVjyu��z�boqx�l�}k�"lnm�oqp"��j�w~p oýrV�zl������Um�i­pkw~pJ���toqpV�zl�iV�zpk|s�oqrV�z}Vl��Yjq�ì|k�5u�jyoql�}ks�jyu�jypfm�ots�w~}VpVrvm�ikj���jyxzx-l�}ks�{Foq��j u�jyu��Vs�oqpVj\��

¦¯_a^ �»«�f7d���cwgatÔÖ3b�¨nf�oQcei�h� ntazÏÒ c¯jmf%gazºtal�^2x��ü jbs�j,�%j,rVjyl���s��U�.j�oSi����.jbs��.w~xz�z��u�wJrkjyxh{Ywqs@��ikjyu�wqm�oc¦J�zl��zp�w~pkjDln�"oq��j,rV��u�jypJ�l��Uw~p oqpVr;w~}km�xz��pkj�m�iVj��"oql��z���Vs�wq�.jbs�m��Ujyl�oqpVr�s�jyl�}VxUm�l��/iV����i�ots�j�¨Jpkw��/p;mnw�rVotmnjq�g(ikj�w~pkj¢rV�zu�jyp"l��Uw~pVoqx.u�wJrkjyx��zl/ot�V�"xz�z�bot�NxUj�mnw_rVotm�o@�/iV�z�i�iVoy�qj��.jbjyp,��w~xzxUjy��mnjyr�zp_i"�U|~iVxU��ln�Ju�u�jbmns����Të�w~pkj���r"�zu�jypVl��zw~pVoqxYìMj�¦J�.jbs��zu�jyp�m�lM�Yjq�ì|k�5¯hikjyp-�J®kwqs�r�¯h}"u��u��zpk|~lÀ);@A,��qwqs»´/�U�qjbs�w�);<�@A,Y�� þ,wqs�jbw��qjbs��fm�ikjhlnm�}Vrk� wq{Vm�ikj5w~pkj���rV�zu�jyp"l��Uw~pVoqx�u�wXrVjyx�Vs�w��J�zrkjyl5|qwXwXrv�zp"l��U|~i�m�l%�zp�mnw�m�ikjMoqpVoqxU�Xm��z�boqx#mnw�w~x�lb�J�/iV�z�i_ots�jM�zu��#wqs�m�oqpfmh�zp]D wqsK�rV�zu�jyp"l��Uw~pVlhoql��5jyxzx´��«pDj�¦J�#jbs�zu�jyp�m�l(wq{%ªXw~xzxPoqpVr ¬ jyl�lnjyxzl�);:A+�,��#®B�zl�ikjbs��4 À����O�) ù +�,��.oqpVrDwqm�ikjbs�l��Um

m�}ks�pVl5w~}kmhm�iVotmhm�ikjMln�.jbjyr�oqpVr�m�ikj�m�}ks�pV�zpV| s�otmnjylhwq{W�zpVrV�U�J�zrV}"oqxzl9rkw�pkwqm�rkjb�.jypVrw~pVxU��w~p_m�ikj���w~pV��jyp�mns�otm��Uw~p�wq{-m�ikj�j�¦Jmnjbs�pVoqx#l��U|~pVoqx������$�*�-�B�"}Vm%oqxzlnw�w~p@mnjyu��#wqsoqxoqpVr�ln�"otm���oqx-�cots�zotm��Uw~pVl�� � �����*�.��oqpVr�� � ���$�*�-��%g(iVj?��w~xzrVlnmnjy��pJ�¤©�oq�Mu�wJrkjyx�� ù �h��ljyoql��zxU��j�¦Xmnjyp"rkjyr�mnw@�zp"�bxz}Vrkj���ikjyu�wqm�oc¦J�zl�j��.jy��m�l��

} �� � �'{B������� � ��� � �Ô}��9� � � ��|w�h������� � ��� � �Ô}��Ï�3|¯��������� � ��� � �Ô}w�n�} �� � �'{B������� � ��� � �Ô} � � � � | � ������� � ��� � �Ô} � �~| � ������� � ��� � �Ô} � �¢�� � � ��� ��� �.á9�����*} � �Ø} � �ª��¢Ï°èC�}��h�"C��� �m��} �Ü � ���"C��� �J��� Ü �� ù Cf�

G

Page 8: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

�«p�m�iV��lWpVwqm�otm��Uw~p�m�ikjhs�otmnjyl»|��Dots�j�m�}kspV�zpk|�s�otmnjylb�f��ikjbs�jyoql9�zp�� ù �p|��zl9o�lnmnwq�V�"��pk|s�otmnjMoqpVr�jyoq��i�rV�Us�jy��m��Uw~p@���zxzxV�.j���ikw~l�jyp_���Um�i@�"s�wq�"ot�"�zx��Um¤��wq{ ù ��DX�5g(ikj�{Y}Vp"��m��Uw~pá9�����*}��¥�Ï}e�³�Brkjyl���s��U�.jylB�Vs�wJrV}V��m��Uw~p�oqp"r�rkjy�boy��wq{-m�iVj(j�¦XmnjbspVoqx"l��z|~pVoqx´�BªJ�Xlnmnjyu� ù Cf�(s�jy �}V�zs�jyl(oqrVrV�zm��Uw~pVoqx�oql�l�}Vu��Vm��Uw~p"l�mnw@jypVl�}ks�j o@�%jyxzx³rkj�="pkjyr�¯hoq}V�if���Vs�wq�k�xUjyu��#�/iV�z�iDrkwXjyl�pkwqm�oqxzxUw��'{Ywqs��Noq��¨X�5ots�r�rV�U�.}Vl��Uw~p �Y{Ywqs�j�¦koqu��"xzj�m�iks�w~}k|~i,m�iVjrkjb�.jypVrkjypV��j�w~p�� � �� ¥ �.w~l�l��U�NxUj�lnjbm/wq{Wl�}V�@�b�Ujyp�m(��w~pVrV�Um��Uw~p"l5��lh|~�U�qjyp��zpB);K�:A,�wqs�zpB) + ù ,��ªX�.jy�b�zoqxP�boql�jyl/wq{/� ù Cf�/�5jbs�j�lnm�}"rV�Ujyr�wqs�ot�V�"xz�zjyr����,ªXjb|qjyxÀ);:�KA,��r´/�U�qjbs�w��� ����PO);<�@A,�����s�jbjyp��.jbs�|@oqpVr ¥ xUm¥);D ù ,��"oqpVr,¯hikjyp3�� J����O¿);@J��\I,���«pN);K�\�,/xUwX�boqxhoqpVr�|~xzwq�"oqx5j�¦J�zl�mnjypV��j�wq{�lnw~x�}km��Uw~pVl�iVoql��.jbjyp3�Vs�w\�qjypÿ{Hwqs�m�iVj

�boqlnj�wq{M��w~pVlnm�oqp�m@ln�.jbjyrÿoqpVr�{Ywqs�m�}ks�p"�zpk|;s�otmnjyl@rkjb�.jypVr"�zpk|;w~p�� oqpVr � � �ÿ��p);K�:A,³|~xUwq�Noqx��zp�m��zu�jMj�¦J�zl�mnjypV��j {Ywqs�{º�y{9��� �hiVoql/�#jbjyp,l�ikw���p-�V�/ikjbs�jMm�ikj�l��z|~pVoqxrV�zlnmns��z�"}km��Uw~p �hoql�oql�l�}Vu�jyr mnw;�.j��zpý �}"oql�����jy X}V�zxz�U�"s��z}Vu �"¢.� Cf�� g(ikj_s�jyl�}Vxzm�lwq{¥);K�\A,5oqpVry);K�:A,5i"oy�qj��#jbjyp j�¦Xmnjyp"rkjyr s�jy��jypfm�xU�S��pF) + ù ,%mnw��zpV�bx�}Vrkj!� � oqpVr"� ���rkjb�.jypVrkjypV��j��zpvm�iVjMm�}ks�pV�zpV|¢sotmnjyl/oqpVrÝ¢$#�ÆC�{Hwqs/m�ikj��boqlnj�l�m�}VrV�Ujyrv�zpB);K�:�,��¦¯_'¦ §T¨mf&%�hrdRh���cwgÔtaÖB©�tao�t iÝtal�^2x���«p£m�iV��l�lnjy��m��Uw~p£�5j s�jy�boqxzxMlnw~u�j wq{¢m�ikj s�jyl�}VxUm�l�{Hs�w~u );K�\A,��M�zpR�"ots�m��z�b}Vx�ots�m�iVjl��boqxz�zpk| oqpVoqxz�Xl���l��/iV�z�i3xzjyoqrVl�mnwSo �"otsot�#w~x��z�_x��zu��Umb�3�5oql��z�boqxzxU�ým�ikjbs�j,ots�j�m��%wj��#jy��m�l�xUjyoqrV�zpV|;mnwý��ikjyu�wqm�oq��m����,ot|q|qs�jb|~otm��Uw~p�� M �zm�ikjbs��"ots�m��z�bxzjyl�l�xUw�� rkw��/p otmiV�U|~i���w~pV��jypfmnsotm��Uw~pvxUjb�qjyxzlb�Xwqshm�ikjb��m�}ks�p_xUjyl�l%wq{Hmnjypv�U{-m�iVjb��u�w��qj�}k��o |qsoqrV�Ujyp�mb�M �Um�ikjbs/j��#jy��m��zl�l�}k�@�b�zjypfm�mnw��"s�wXr"}V��jMot|q|qs�jb|~otm��Uw~p-�gPw�rkw�m�ikj�{Ywqs�u�oqxBoqln�Ju��Vmnwqm��z�bl��5j¢s�jb{Hwqs�u¢}Vxzotmnj�� ù Cf���zpDmnjbs�u�l�wq{�m�iVj�mnwqm�oqx

rkjypVl��Um��T}Ï�y} � �3} � oqpVr�m�ikj�r"���#jbs�jypV��j('E�y} � �~} � �} � � �'{�'~� � � C�' � � �'{�}.� � � ����|w����|¯�P�Ô}�� ��|w�Ï�3|¯�³�)'2�¢�� � � ��� ��� �.á9�����*}-�$� � ùqù �

ªJ�zu���xzots mnw�m�ikj�oqpVoqxU�Jl��zl wq{Pm�ikjMxz�zpkjyots5u�wXrVjyx³� ù ��k�5j��boqpvrkjbs�U�qj�o�mnjyxUjb|qs�ot�"ijy �}Votm��zw~p-�

} �P� �+*>} � � �'{B�'{�}.� � � � � �'{9��| � �~| � � � �,* � 'E��C� � ù D~��/�Um�i�oqp,oq}k¦J�zx��zots���{F}VpV��m��Uw~p

*³���w�*���À�þ�y| � ������� � ��� � �w�3| � ���¯��� � ��� � �¯� { � ������� � ��� � �{B������� � ��� � � � � ù K~�¬ j�s�jy X}V�Us�j�m�iVotm-*;rkwXjyl/pVwqm�rkjb�.jypVr�j�¦J�"xz���b�Um�xU�@w~pDm�ikj¢ln�"oq��j �tots��zot�NxUjE�P��g(i"�zls�jylnmns��z��m��zw~p�w~pÏ|��J�6{���lhmns�}kjM{Ywqs/u�oqp��vot�"�"xz�z�botm��Uw~p"lb� ¬ j�|~�U�qj�j�¦koqu��"xUjyl/xzotmnjbsy�

\

Page 9: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

¬ j��zp�mns�wJrV}V��jMo�l�u@oqxzx-l��boqxz�zpV|¢�Nots�oqu�jbmnjbs�£��èC�oql{Ý� { Ü£ � oqpVr | � � | �Ü£ [ � � ù +��

�/ikjbs�j1{ Ü oqpVr�| �Ü ots�j_wq{(wqs�rkjbs ù �/�Um�i s�jyln�.jy��m mnw�£~����pým�iV�zl�l��boqx��zpk|��5jT=NpVrm�iVotm *P�����n� * Ü ���n�£ [ � �/�Um�i * Ü �y| �Ü �3| �Ü �B£ [ { Ü�. �{ Ü � ù <~�oqpVr�{Ywqs�£¿íîC��%j wq�Vm�oq�zp

�Ó� x��zu/10 Ü { [*P����� � { [Ü| �Ü �3| �Ü � � ù :~���{-�5j�l��boqxUj�m�iVj���wqs�s�jyl��#w~p"rV�zpk|�mnjyxzjb|qs�ot�"i_jy �}Votm��zw~p_oq�b��wqs�r"�zpk|~xU��m�ikjyp_�5j�wq�Vm�oq��p{Hwqs�£�í C�m�iVj��"ots�ot�.w~xz�z��xz�zu@�Um%jy X}Votm��Uw~p

} � � �"�T} � �~}32(� � � � ù Gq��/�Um�i¢o�rV�U�.}Vl��Uw~p���wXjb�@�b�Ujyp�mW|~�z�qjyp �X�_� ù :~�³oqpVr¢m�ikj���������R 8�R�4 "� �S-����P�4���" "!42 |~�U�qjyp��� 2è�v� { Ü| �Ü �3| �Ü

5 { Ü�. � � xz�zu/60 Ü ù£ ��| �Ü �~| �Ü �87º� � ù \~�¥ ln�.jy�b�zoqx#�boqlnj�wq{�m�iV�zl%s�jyxzotm��Uw~p_�hoql5rVjbs��U�qjyr�oqpVr_}"lnjyrv�zp1´��U�qjbs�wÝ�� n���PO¿);<�@A,.oqp"rªXjb|qjyxJ);:�KA,����{323iVoqlWm�ikj�{Ywqs�u92M������� � �n� � ��� �1� � m�ikjyp�� ù Gq�B�zlWm�ikj��bxzoql�l��z�boqxJ��ikjyu�wqm�oc¦k�zlu�wXrVjyx-�/�Um�iB��������� 8�R�/ "�"�?�4���>���� "��-/�" "! � ������g(ikj �ikjyu�wqm�oq��m��z�¢�qjyxUwJ�b�Um�� � ù \~�(��w~pVl��zlnm�l(wq{9m��%w�mnjbs�u�l/�/iV�z�i,�boqpD�.j��zpfmnjbs��

�Vs�jbmnjyr��zp"rkjb�.jypVrkjyp�m�xU���ù � ¥ l�l�}Vu�j�{Ywqs(pkw\� m�iVotmÀ{Ï�y{B��� �À°FC�oqpVrvm�iVotm�m�ikj�rV�U�#jbs�jyp"��jª| �Ü �Ì| �Ü ��lwq{.wqs�rkjbs�£;:k��{HwqsÀ£ªíîCJ���/�zm�i@lnw~u�j=<Ý� ù � g/ikjyp�|w����|¯�5��>��"£?:�� [ �9oqp"r2è��� { Ü| �Ü �3| �Ü {�ÇÜ � � � � ù @~�g(iVj�x��zu��Um��zpV|�jy X}Votm��Uw~p�{Ywqs/�ikjyu�w~lnjypVl��Um��U�qj u�w\�qjyu�jyp�m¢� ù Cf�hs�jyoqrVl

} � �A@ { Ü| �Ü �3| �ÜCB { Ü } � ��{�ÇÜ � � }�DFE � � � D�Cf���p �boqlnj�m�iVotm¢m�ikj@�.wq�"}"xzotm��Uw~p l�xUw\�/l¢rkw��/p3otm�i"�U|~i ��w~pV��jyp�mns�otm��Uw~p"l�wq{G�h�{ Ç ��� �IHîCJ�Mo­pVjbmÏ�Vw��&wq{¢m�ikj �.wq�"}Vx�otm��Uw~p¡��p£m�ikjSrV�Us�jy��m��zw~p£wq{¢iV�z|~ikjbs��w~pV��jyp�mns�otm��Uw~p"l(wq{J��wJ�b�b}ks�lb� g(i"�zlÀ�Vw��1lnm�ot�"�zxz�z�bjyl�ot|q|qs�jb|~otm��Uw~pVlb��«{À{ Ç ��� �T��CJ� �´�ìjq� �U{��"ots�m����bxUjyl�oq�b��jyxUjbs�otmnj�otm�iV�U|~i3��w~pV��jyp�mns�otm��Uw~pVl�wq{-�h�m�ikjyp�m�ikjM�.wq�"}Vxzotm��Uw~p��/��xzx.ln�Vs�jyoqr�w~}Vmb�

@

Page 10: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

DX�/�«{B�%j�oql�l�}Vu�j�m�iVotm�{Ì��KMLON�P4���"m�iVjyp,m�ikj?=Vs�lnm/mnjbs�u ��pý� ù \~�(�coqpV��l�ikjyl/oqp"rm�ikj���ikjyu�wqm�oq��m������qjyxzwX�b�Um����zl |~�U�qjyp@����m�ikj�rV���.jbs�jypV��j(�zp�m�ikj/m�}kspV�zpk|�s�otmnjylb�¬ j oql�l�}Vu�jMm�iVotm/m�iV�zl(rV�U�#jbs�jyp"��j��boqp,�.jMj�¦X�"oqp"rkjyr�oql

| � ��| � �y£ � K�Q K � Q [ ��/�zm�i�{F}VpV��m��Uw~pVl Q K ������� � ��� � �$� Q [ ������� � ��� � �� ü jypV��j�m�ikj¢s�jyl��boqxUjyr;m�}ks�p"�zpk|s�otmnjyl%l�otm��zln{Y�¥| �Ü ��| �Ü ��£ Q K �]£ [ Q [ �9oqpVr�m�ikj(�ikjyu�wqm�oq��m��z���qjyxUwJ�b�Um¤�¢s�jyoqrVl2F��� { Ü| �Ü �3| �Ü Q K �®kwqs�o�xz�zpkjyots�{F}VpV��m��zw~p Q K � Q Ü �����1� � o�§�©¢ª�u�wXrVjyx-�zl�wq�Vm�oq�zpVjyr

} � � @ { [Ü| �Ü �Ø| �Ü } � �{ Ü| �Ü �3| �Ü Q Ü ��� �1� � } E � � � D ù �

g(iVj�l��U|~pvwq{Bm�ikj rV�U�#jbs�jyp"��j��zpvm�}ks�pV��pk|�s�otmnjyl/rkjbmnjbsu��zpkjyl�m�iVj�ot|q|qs�jb|~otm��Uw~p�.jyiVo��X�Uwqs���«{R� � � C oqpVr Q Ü H CJ�5m�iVjyp­m�ikj��Nots�m�wq{�m�ikj��.wq�"}Vx�otm��Uw~pÿu�w��J�zpk|S}V�m�ikj�|qs�oqr"�Ujypfmb�km�}Vs�pVlhxUjyl�lhwq{Ymnjyp�m�iVoqpvm�ikjM�.wq�"}"xzotm��Uw~p�u�w��J�zpk|�rkw\�/pf�hots�rVly�g(i"�zl5�#jyi"oy�J�Uwqs/iVoql��.jbjypvwq�"l�jbs��qjyr,�zpv�Noq��mnjbs��zoJ��«{�� � �õC;oqpVr Q Ü �õCJ�9m�iVjypSm�ikj_�#wq�N}Vxzotm��Uw~pSu�w��J�zpk|,}k�ým�ikj@|qsoqrV�Ujyp�mb�m�}kspVl�u�wqs�j_wq{Ymnjyp�m�iVoqp3m�ikj��.wq�"}Vxzotm��Uw~pýu�w��J�zpk|�rkw\�/pf�hots�rVly�Sªk}V��i3�.j��iVo��X�zwqs(�%w~}"xzr�rkjylnm�ot�"��xz�U�bjMot|q|qs�jb|~otm��Uw~p-�

ü jyp"��j m¤�5wÿj��#jy��m�l�xzjyoqr¡mnwÿ�.w~l��zm��U�qj,�"��oql�� ¬ ikjyp£u�w��J�zpk|ÿ}k�X�hots�r¡o�l��z|t�pVoqxP|qsoqrV�Ujyp�mb�#�zp"rV�U�J�zrV}Voqxzl5l�xUw\�'rVw��/p-�#m�ikjb��xUw\�%jbs�m�ikjy�zs/m�}ks�p"�zpk|�s�otmnjq�#wqs��.wqm�i-�g(ikjylnj�j��.jy��m�l/iVo��qj�oqxzlnw��.jbjyp_{Yw~}VpVr��X�vªJ�iVpV�Umn�bjbs¥);:�C�,-��p�o�l���u��zxzots5w~pkj��¤rV�zu�jypJ�l��Uw~pVoqx.i����.jbs��.w~xz�z�/u�wXrVjyx"{Ywqs��"oq��mnjbs��zoqx.u�w\�qjyu�jyp�mb�V�/ikjbs�j�u�jyu�wqs��@j��.jy��m�l(iVoy�qj�#jbjyp,�zpV�bxz}VrVjyr-�

S *D�-�(���#���/�#!'6 ��)�%���

g(ikjbs�j@��l�o�s��z�iSxz�Umnjbsotm�}ks�j�w~pSmnsoqpVln�.wqs�m�u�wJrkjyxzl�ot�"�"xz�Ujyr;mnw,�.wq�"}"xzotm��Uw~pVlb� ¬ js�jb{Hjbs�mnw�m�iVj�ots�m��z�bxUjyl ¬ � ¥ xUm�) ù �7DA,���L�m�i"u�jbsy�>ú�}Vpf�Notshoqp"r ¥ xUm�);<�KA,��N¯hiVjyp~�4 J���PO);@A,��»ú�����¨J�zpVlnw~p3oqpVr g³s�oqpV X}V�zx�xUw�) ù DA,��nú��z��¨J�zpVl�w~pQ) ùqù ,�� oqpVr ü ��xzxUjypSoqpVr.L�m�iVu�jbs);KA+V��<A+�,���«p.);KA+V�e<A+�,9o�|qjypkjbs�oqxBm�ikjbwqs��,iVoql��.jbjypDrVjb�qjyxUwq�.jyrDmnwvwq�Vm�oq��pDm�iVj¢�Nots�ot�.w~xz�z�

xz�zu��Um��FrV���-}Vl��zw~p�xz��u��Um��{Hwqs�o�|qjypVjbs�oqxkmns�oqpVl��#wqs�m³jy X}Votm��Uw~p���iV�z��i�rkjyl���s��U�.jylBu�w��qj��u�jypfm³wq{k�.wq�"}Vxzotm��Uw~p"lb� ¬ j5�Vs�jylnjyp�m�m�ikj%�"oql��z�9s�jyl�}VxUmP��pMm�ikj�{Yw~xzxUw\�/�zpk|(l�}k�"lnjy��m��Uw~p"lb�

ù C

Page 11: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

�«p3m�ikjv�boqlnjvwq{/�ikjyu�w~lnjypVl��Um��U�qj�u�w��qjyu�jypfm��zp£��+�� m�ikj�m�}ks�pV�zpV|�s�otmnj]|Roqp"rm�ikj,�qjyxUwJ�b�Um���r"�zlnmns��U�N}km��Uw~p3¨qjbs�pkjyxÀÅ����%�*� Ç ��rkjb�.jypVrÿw~p­m�iVj,l��U|~pVoqx/rV�zlnmns��z�"}km��Uw~p�����$�*�-��Nw~p;�Um�l/|qs�oqrV�Ujyp�mªÃ!�(���$�*�-��"wqs�w~p,wqm�ikjbs��Vs�wq�.jbs�m��zjyl�wq{T� �Yjq�ì|k�/pVw~pJ�¤xUwJ�boqxrkjb�.jypVrkjypV��j��boqp��.j��zp"�bxz}VrkjyrN��

|º��|�������Ã!���������×�$� Å¢���%�*�&ÇH�»�QÅ¢�����*��Ç ������Ã!�¯�������×� � D�D~�ѯ_a^ U]h`zAtÔÖV~z�z�®moy«niRt8cplmzXW�§Ì^�Y�xZW�§1ã[Y¬ jD��w~pVl���rkjbsD��+���w~p \ �&�H´ µ � ¬ j,oql�l�}Vu�j�m�iVotm]¶ · �H´ µ �zl@��w~u��"oq��m_oqp"rln�Xu@u�jbmns��z��l�}V��i,m�iVotmS�ϲ̶ �zu��"xz�Ujyl¿���ϲ̶��^]�jbmG_ rkjypkwqmnj m�iVj ��w~pkj wq{ pVw~pJ�pkjb|~otm��U�qj {F}VpV��m��zw~pVl��zpa` [ �8¶��� ¬ j rVj�="pkjMm�ikj�{Yw~xzxUw\�/�zpk|¢wq�.jbs�otmnwqs�l/w~pb` [ �8¶��$�c ¹W���J� � Ä�ß�Å¢�����*� Ç �ȹW��� Ç �ZÉ�� Ç � ced ¹W���J�J�ÆÄRß�Å���� Ç �*�J�ȹW��� Ç �ZÉR� Ç �f � ��|��hg¥� c �$��/ikjbs�j(g�rkjypkwqmnjyl�m�iVj �zrkjyp�m��Um¤�q�%®kwqs/m�ikjM¨qjbspkjyxrÅ<�5j oql�l�}"u�j��i > ûkj Å¢�����*� Ç ��°èC� Ê Å¢���%�*� Ç �ZÉR��� ù � oqpVr Ê�Ê Å [ ���%�*� Ç �ZÉR� Ç É��lHnmR�i > ý�j g(ikjbs�j5j�¦k�zlnm�lBlnw~u�jÀ} Ü ²o_¡���Um�i�} Ü #Ð�CJ�flnw~u�jh��pfmnjb|qjbsJp oqpVr�o���w~p"lnm�oqpfmé �èC�l�}V�i�m�iVotm({Ywqs/oqxzx �����*� Ç ��²º¶rq5¶} Ü ���J�4s Å-t���� Ç �*�J�"s é } Ü ���k�$�

�/iVjbs�j�m�ikjup,��m�i��Umnjbs�otmnj�wq{wÅ1��lÅ t ���%�*�&ÇH�J�þ�NÄâ������ÄÆÅ¢���%� æ�K �aÅ¢� æ�K � æ([ ��Â�Â�Â�Å¢� æ t � K �*�&Ç��ZÉ æ�K ������É æ t � K �i > ��jwv c v�x K8y{z H ù �-�/ikjbs�jw| ù;}6~ rkjypkwqmnjyl�m�iVj¢wqs�m�ikwq|qw~pVoqx9��w~u��"xzjyu�jypfm�wq{�m�iVjl�}V�"ln�"oq��j�| ù;} ·�` [ �8¶��%wq{W{F}VpV��m��zw~pVl���w~pVlnm�oqp�m/�zp]�N�i > �3j Ê ß Å¢�����*� Ç �ZÉR� Ç � ù �� pVrVjbs¢m�iVjylnj�oql�l�}Vu��"m��Uw~pVl�m�iVj�m�}Vs�pV�zpk|,wq�#jbsotmnwqs f iVoql¢m�ikjv{Hw~x�xUw��/��pk|D©Ms�jy�zpJ�´/}km�u�oqp��Vs�wq�.jbs�m��Ujyl��

>��³G�²NZ\G�� û�� ���$(�1�¢�Fg ù �¤��Fg�+���Ol�w����� OªCº�È�¥�1�$����#��Y�����P[&���7-A����(%���6� f�� �" ��5����[2������(���4 "� ���b�W���J�mÐ ù O� Oa�w���*�1�çô��'�� ���������� ��2�ç[�������������������#�A���� "� ���$` [ �8¶��m�r| ù;}�� | ù;}6~ �����=�����������y²X| ù;}6~ � �¥���-��Ä�� f �JÉ��lsN�G� [ v � v [� éM� ß�� � � �� �� � [ Ðy|�� ù � v c v�x K8y{z �$�

ùqù

Page 12: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

� O����R��������[2���7-�����(7��� #��CÌ���� "�'�'ÿÀ������DA|,H��S�-�Xsõ��� [ HNC��������5 ����ç���È����]�� ��>��À#���$�" "�'-������P[&�����$(���/ "�"����O� O v f v � � � é�� ß�� . � é�� ß��¡� sFDA|mO¢ O f ç���� "$� �4 a� �] 8�l| ù;} ~ ·,` [ �8¶��ª�2�����T�þ�����ç��¥���7-�����4�¤£ � �� ��Ý����$�v £ v � � x K8y z . x K8y z � s ù� [ �g(iV�zl%m�ikjbwqs�jyu��boqp��.j��Vs�w��qjypvoqxUw~pk|¢m�ikj�{Hw~xzxzw��/�zpV|�xz�zpVjylb��lnjbj�);KA+�,�� ®ks�w~u&�Fg ù �%�Um{Hw~xzxzw��/l5m�iVotm c oqp"r c d ots�jM��w~u��"oq��m ü �zxU�.jbs�m��«ªJ�iVu��zrkm�wq�#jbsotmnwqs�lb� ¥ l�l�}Vu��"m��Uw~p�Fg�D~��jypVl�}Vs�jyl�m�iVotm c �zl¿} Ü ���#w~l��Um��U�qj��zp;m�ikj�lnjypVl�j¢wq{h©Ms�oql�pkw~lnjyxzln¨J�z�À) +�:A,��Pikjyp"��jm�ikjbs�j�j�¦k�zlnm�l�o�}VpV�z X}kj��.w~l��Um��U�qjMjy�z|qjypk{Y}"pV��m��Uw~p��W���J�J� ù �h®ks�w~u �Fg�+��5�5j wq�Vm�oq��pm�ikj�wqs�m�ikwq|qw~pVoqx�rkjy��w~u��.w~l��Um��Uw~p;wq{^` [ �8¶¢���g(ikj�oql�l�}Vu��Vm��Uw~pÿ�Fg�K~���zl�jyl�l�jypfm��zoqx����m(jyp"l�}ks�jylhm�iVotm/m�ikj�m�}ks�p"�zpk|¢wq�#jbsotmnwqs��zl�rV��l�l��U�"otm��z�qjq�¥ l�l�}Vu��Vm��zw~p��Fg�D~���zl�|qjypkjbs�oqxBjypVw~}k|~i mnw��zpV�bx�}Vrkj¢m�}ks�pJ�¤oqpk|~xzj�r"�zlnmns��U�N}km��Uw~pVlb�

�/iV�z�i�ots�j��bjbs�wM{Ywqs�xzots�|qjbs9m�}kspJ�¤oqpk|~xUjylb�W�«{��~{Ywqs9j�¦koqu��"xUjq�f��pVrV�U�J�zrV}Voqx�l���ikwXw~lnj/pVjb��qjyxUwX�b�zm��Ujyl��zp�oqp_ots���wq{r+�<Z¥���w~u��Nots�jyr�mnw�m�iVj(�Vs�jb�J�Uw~}VlB�qjyxUwJ�b�Um��q��m�ikjyp@m�ikj(�Umnjbs�otmnjcu¦ �5w~}Vxzrv�.j�} Ü �.w~l��Um��z�qjq�ѯ_'¦ %�h�d�h���cpgataÖ"§¯Öh`gatÔlnäL�p@m����N�z�boqx"j�¦J�.jbs��zu�jyp�m�oqxNm��zu�j/�.jbs��zwXrVl�� K�iÏ�SO������þ�z�9�Noq��mnjbs��zo¢l�ikw�� u�oqp���m�}ks�p"l�#jbs�}"pV�Um�wq{%m���u�jq� ¬ j@�boqp �zrVjypfm��U{Y��m�iks�jbj�m��zu�jyl�l��boqxzjylMm�ikjÏ�1�ç���Ø�(�3 "���1��wq{ot�#w~}Vm ù lnjy�t�¨���� �� "W6 "�'�1�¥�������Y��wq{Bot�.w~}km ù C�C�m�}kspVl�oqpVr�m�ikj!��� üS(2���"���%WZ "���1���������Y�wq{hot�.w~}km ù C�C�C�CD�zpVrV�z�X�zr"}Voqx�m�}kspVlb��gPw,s�j��Vjy��mMm�ikjylnj_l��boqxz�zpk|~lM��p�m�ikj@u�wJrkjyxB�5j��ikwXw~lnj�m�ikj�#��6��©������"�?���������'�2[/���Um�i@o�l�u�oqxzx�£��"£���C�HC ù {Hwqs5m�ikj�j�¦Joqu��NxUj/ot�.w��qj\��

¢1��£ [ � oqpVr ¤E�Æ£4�w� � D�K~�¬ j s�jyl��boqxzj�jy �}"otm��Uw~p ��+���"��iV�z��iv|~�z�qjyl

£ [ ¹�ª��3£4��ÂIÃe«a¹]� f ¹w� � DA+��®kwqsª÷Ï�QD��%j���w~p"l��zrkjbs/oqp�j�¦X�"oqp"l��Uw~pvwq{�¹w�

¹W�"¢�*¤�*�J�m� ¬­ê¯® Ü ¹ ê �"¢�*¤�*�k�Z£ ê �5¹ ¬ � K �"¢�*¤�*�J�Z£ ¬ � K ���{W�%j���w~xzxUjy��m(mnjbs�u@l�wq{Bjy �}Voqx�wqs�rkjbs(�zpÝ£ �%j |qjbm��

£ Ü � C?� f ¹ Ü �£ K � ��Â/ê¹ Ü � f ¹ K �£ [ � ¹ Ü ª ����Â/ê¹ K � f ¹ [ � � D�<~�ù D

Page 13: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

�/ikjbs�j;m�ikj l�}k�"l���s��U�VmT¤ w~p£m�ikj�pVot�"xzoýwq�.jbs�otmnwqs�iVoql��.jbjypRw~u��Umnmnjyr-� ¬ jS}"lnjm�ikj;l��#jy��mnsoqx/�Vs�wq�#jbs�m��Ujyl�wq{ f �Fg(iVjbwqs�jyu ù ��1g(ikjDwqs�rkjbs�w~pkj;jy �}Votm��zw~p |~�U�qjyl¹ Ü �õ¹ Ü �"¢�*¤f�� ü jypV��j�¹ Ü ��l¢��pVrkjb�.jypVrkjyp�m�wq{S�F²N¶��ýªJ�zpV��j϶ �zl¢oql�l�}"u�jyr3mnw�#j�ln�Ju�u�jbmns����t�Xm�iVj�l�w~xU�cot�N�zxz�Um�����w~pVrV�Um��Uw~p Ê �¿Â�ê¹ Ü ÉR����C¢wq{Pm�ikj�£ K ��jy X}Votm��Uw~p���ll�otm��zl*=Vjyr�� ü jypV��j`¹ K � £,����Âþê¹ Ü ��c�/ikjbs�j�£<�zl-m�ikj9�"l�jy}Vrkwt�¤�zp��qjbs�lnj9wq{ f s�jylnmns�����mnjyrmnw m�ikj�ln�"oq��j�| ù;} ~ �Xoql�rkj�=Npkjyr��zp�g/ikjbwqs�jyu ù � g/ikj(lnw~xU�tot�"�zxz�zm¤����w~p"rV�Um��Uw~p�{Ywqs�m�iVj£ [ ��jy X}Votm��Uw~p�s�jyoqrVl ÄRߪ¹ Ü ª �3�?Â/ÿ¹ K ÉR����C�� l���pk| m�ikjDs�jb�Vs�jyl�jypfm�otm��Uw~pRwq{�¹ K {Hs�w~u ot�.w��qjq��oýr"���.}"l��Uw~p�jy X}Votm��Uw~pR{Hwqs?¹ Ü ��lwq�Vm�oq�zpkjyr

¹ Ü�. ª �ÆÃ�Â4�]ê¹ Ü � ¹ Ü ��¤�çCf�m�uÄ�ߪ¹W��¤�*�%�çCf�ZÉ��%� � D�:~��/�Um�i�rV�U�.}Vl��Uw~p�mnjypVlnwqs ��Ð�� ù° Ä�ß���£���±?ÉR��� � DRGq�

g(ikjh�Vs�wJ��jyrV}ks�j5�boqp��.j5��w~p�m��zpX}kjyr�mnwMiV�U|~ikjbsBwqs�rkjbslW��p?£�oqpVr-�qoql9iVoqlB�.jbjyp�l�ikw���p�zpB);KA+R,��km�ikj s�jyl��zrV}"}Vu wq{³m�iV��l�ot�V�Vs�w�¦J�zu@otm��Uw~p��boqp��.j���w~p�mns�w~xzxUjyr`�>��³G�²NZ\G�� ý³² !�������(7�/ "�"�����ª����9�(%���������6���$(���/ "�"���>�%¹ à �$���w©$�������>�� "�(7�4 a�F���$(7��� ��2�� � ��>���$(���R' ¬ �þ�9´ ¬à ® Ü £ à ¹ à �2��-��1 ���¤�������P� � �'�2[�#�*�$#>��� "!�µe¶e��1���R���l·Ø�uC ����ç���'�E�]�$���>�� 8���% �¸T¹1�èC5�$(7���Ý ��2�R v ¹W���p���P�*�n�n�' ¬ ���w���P�*��� v � é�� ߨ� s ¸T¹J£ ¬ � K�����������· � £ [ H.�^H m �����Ý�ç�������̲b\¥Oü jypV��j%m�ikj�oqln�Ju��Vmnwqm���� �.jyiVoy�J�Uwqs�wq{klnw~x�}km��Uw~pVl-wq{���+����zl�rkjyl���s�U�.jyr�����m�ikj�r"���.}"l��Uw~pjy �}Votm��zw~p �zp � D�:~��vg(ikj@�Vs�wXwq{%wq{hm�iV�zl�s�jyl�}VxUm�}"lnjyl�oqp ��pVrV}V��m��Uw~p�ots�|~}"u�jypfmb����p�"ots�m��z�b}Vx�ots��Vs�wq�.jbs�m¤� �Fg�K~�h��l��zu��.wqs�m�oqp�m�mnw@l�ikw\� m�iVotm(m�ikj�xz�zu��Um���pk| jy X}Votm��Uw~pD�zp� D�:~���zl��Nots�ot�.w~xz�z�t�ѯ_'Ñ §T¨mfI��t»º�®mz�tacwl�§�f�lmzAced¼��«pÿm�i"�zl�lnjy��m��Uw~p �%j,rVjbs��U�qj���w~pVrV�zm��Uw~pVl�{Ywqs�m�ikj�rV���-}Vl��zw~pýmnjyp"lnwqs_�zp � DRGq��mnwS�.j�zlnwqmns�wq�"���t�J���ìjq�U�kmnw��.j�o�l��boqxzots(u�}"xUm��U�"xUj�wq{³m�iVj��zrkjyp�m��Um¤�q� ¬ j rVj�="pkj�m�ikjT�çô�#>�ç�/ a�F�-����P�4�4�� "! ½�.���J�nÐuÄ�Å¢�����*��ÇH�Ô�&Ç�É���Ç"� � D�\~���m�{Yw~xzxUw���l/{Ys�w~u �Fg ù �/m�iVotmEÊ ß ½�.���k�ZÉR�5�vCJ� þ,wqs�jbw��qjbsy�W�U{J¶���l�ln�Ju�u�jbmns�z� ���Um�is�jyln�.jy��m(mnwl�J>@�¾N³��km�ikjyp�m�iVjbs�jMj�¦J��lnm�l(o���w~pVl�m�oqpfm-¿ ß l�}"��i�m�iVotm

Ä ß ��� ± ÉR���C¿ ß gù K

Page 14: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

�Y�/iV�z�iR�boqp£�.j lnjbjyp£�X� ot�V�"xz�X�zpV| m�ikj xUjb{Ym�iVoqpVrRl��zrkj;w~pfmnwÿm��%w mnjylnm����qjy��mnwqs�loqpVr£}Vl��zpk|ým�ikj rV�U�qjbs�|qjyp"��j;m�iVjbwqs�jyu_�����pRm�ikj;�boql�j wq{�¶½�ÀPO� µ � K �5j iVoy�qj¿ ß � ° Ü P [ � N9��«pvwqsrkjbshmnw��bx�ots��U{Y��m�ikj�|qjypkjbs�oqx-l��zm�}Votm��Uw~p��%j�{Ywqs�u¢}Vxzotmnj�m�ikj�{Hw~x�xUw��/��pk| m�iks�jbj�Vs�wq�.jbs�m��Ujylb�i�Á ûkj e��w���ç�Ï�'�]���y��� ��2������$�����R©ç�A���È�]Ë? K �������/� Â µ Î3�6���H´�µè��(7���3 ���� J�����ç�R���¼Ã�� ù �������I�1N³ ���Ì���/��6���'���R a�Ì���$#R#����&[��!Ä ê �À¶ í �H´E� Ä ê ���k�T� � ê ��ç����P[&���;�$(���/ "�"���7�?�6� f�� �" ��Ý�$���������3���P[&���7-A���þ(%�-�ϲ �6��DA|»�çCf��Oi�Á ý�j eÅ�w���ç�Ì�'�º�.�����>�� 8���% ª{³²��6� ù � ù �5�$(7���è ��2�� Æ��������R���B�v²â¶ ����çô�#7�ç�4 a� ��-������4�4�� "! ½�#���k�E���� "�'�'ÿ����½�-���J� v � ����� ½�.���J��Â/�Ù ½�.���k� ÙPÙ � Ù �Q{p�i�Á ��j eÇ�w���*�?�'�?�������>�� 8���% »ÉT�èCÝ��(7���Ý ��2�� ¯�Ó��É�g µ O>��³G�²NZ\G�� ��È �4 ��Fg ù �¤��Fg�+����2���É�~�������A����(���� ���� S¶ �È����!A���1�� "�� � � �� ��~ç��W�8#>���4 À 8�Ê�J>��¾N³��O��w���� � �¥�2��-��

��ª ù � Ë�¼ ��ª>D~� �e¼ ��ª>K~�� ���ç�;©�!� �����$���>�� 8���% ��=�>�6{è������ÉB��ç�?*������ a� �Ý�A�J��������� � ��O{5� �¥�3|| � É���� ¿ ß° � � ¿ ß° |�� ù �~{-� �Ì ��Å ���þ���Ý���R "�È��ÿ���� i >ÎÍ j e��w���*���È���]���� "$�PôwÏ ��(>���º ��2�� ½�#���k����ÏN�Ç������������²º¶Ê�m ���������'�p ���*�$�¥�� 8�� a���1���% �����*����9�(��-A���Y���% 8O

g(iV�zl�g(iVjbwqs�jyu��zlh�Vs�w��qjyr,��pB);KA+�,��¥ x�xqlnm�otmnjyu�jyp�m�l³ots�j�mns�}kj �U{&ÅýiVoql-m�iVj ln�Xu@u�jbmns��z�B{Ywqs�u wq{&Å¢���%�*� Ç �»�y�b� Ù ����� Ç Ù �

�Flnjbj oqxzlnw ¥ xUmE) ù ,Y�� ¬ j �/�zx�x"|~�U�qj�oqp�j�¦Joqu��"xUjM{Ywqs/pkw~pJ�¤�zl�wqmns�wq�"�z��rV���-}Vl��zw~p_�zpvm�iVjpkj�¦Xm�lnjy��m��Uw~p-�

ѯ_Yã V�«m«�gataÖh�iRt8cplÆi�cNb]¨nf%oycwzAf%lJzAt8iRt»Ðwf3ÒÓc�Ðpf%oyf%lpi]�jbmTÅ Ü oqpVrF| Ü rkjypkwqmnj;m�}ks�pV�zpV|S¨qjbs�pkjyx�oqpVr¡m�}ks�pV�zpV| s�otmnj;��p ot�"lnjyp"��jDwq{ oqp����ikjyu����boqx�l�}V�"lnm�oqpV��jq� ��p�);<A+�,��5j;l��Xlnmnjyu@otm��z�boqxzxU�ÿlnm�}VrV�ÿ�#jbs�m�}ks��"otm��Uw~p"l��/iV�z�i��w~u�j {Ywqs�u��ikjyu��z�boqx��b}Vjyl�wq{Bm�ikjM{Hwqsu

Å����%�*� Ç �¨Ñ���n�QÅ Ü �����*� Ç �`�Ø£ ¬¤ÒÅ����%�*� Ç ��Ñ���$� |����%��Ñ�%�»�y| Ü �Ø£?Ó Ò|����%�*� Ç ��Ñ�%�$�{Hwqs»÷���C� ù oqpVruÔe� ù �t�/iVjbs�j¼Ñ��rkjypkwqmnjylBrkjb�.jypVrkjypV��j�w~p�m�ikj%{Y}Vp"��m��Uw~pÎ�,oqpVr pVwqmw~pVxU��w~pMm�ikj xUwJ�boqxq�toqxz}kjT�������*�.��cjq�ì|k�WrVjb�#jyp"rkjypV��j9w~p��������*�.�$��ÃÇ�����$�*�-�$��ÊÎ�(���$�*�-�ZÉR�

ù +

Page 15: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

jbm��t� ots�j��zpV�bxz}VrVjyr-��§Wjbs�m�}ks��"otm��Uw~pVl�wq{�iV�U|~iVjbs�wqs�rVjbs�÷��FÔÀ°uD@�/�zx�xPpkwqm�oc�#jy��mMm�iVj�"ots�ot�.w~xz�z��x��zu��Um(jy X}Votm��Uw~p���§Wjbs�m�}ks��"otm��Uw~pVl�wq{ m�ikj�m�}ks�pV�zpV|�s�otmnj�| Ü wq{ wqsrkjbs�w~pVj�hÔ��kCf��rkw�pkwqm�="m��zp�mnw�m�ikj�{Hs�oqu�jb�%wqs�¨�rVjb�qjyxUwq�.jyrDikjbs�jq�h�h}km�m�iVotm��boqlnj¢�boqpD�.jiVoqpVrVxUjyr���p�m�ikj m�iVjbwqs��vwq{Bu�w~u�jypfm��bxUw~l�}ks�j�oql��zx�xz}Vlnmns�otmnjyr���p,ªXjy��m��Uw~pº<X�hg(ikjbs�jq��Um@�zl_l�ikw���pÿm�iVotm_oqx�lnw wqs�rVjbs@w~pkj��.jbs�m�}Vs��"otm��Uw~pVl���pÿm�ikj,m�}Vs�pV�zpk|�s�otmnj�xUjyoqr mnw§�©�ª���m����.j�u�wJrkjyxzly�¬ j�w~u��Um¢m�ikj�u�w~lnm¢|qjypkjbs�oqxh{Ywqs�u¢}Vxzotm��Uw~pVl¢oql�lnm�otmnjyrÿ�zpÆ);<A+�,�oqpVr5¾n}"lnm�|~�z�qj

lnw~u�j��zxzxz}"lnmns�otm��U�qj�j�¦Joqu��NxUjylb�.�/ikjbs�j¢m�ikj¢�"ots�ot�.w~xz�z��l��boqxz�zpV|�ot�V�Nxz�Ujylb��®kwqs�oqxzxWj�¦X�oqu��"xUjyl��5j�s�jyl�mns��z��m�mnw�=k¦Jjyr�ln�.jbjyr Ù � Ù ��P��$¶v��PO� µ � K oqpVr ° � Ù ¶ Ù �Õ-Ö�L��X×BEYG û e ¬ j�oql�l�}Vu�j�m�iVotm�m�ikj��"s�wq�"ot�"�zx��Um¤�_wq{�o_�iVoqpk|qj¢wq{ �qjyxUwJ�b�Um¤�5� Çmnw���rVjb�#jyp"rVlhw~p�m�ikj�oqpk|~xzj��.jbm¤�5jbjyp�m�ikjylnj m¤�5w��qjyxUwJ�b�Um��Ujylb�

Å K ���%�*� Ç �»� ù° 5 ù �ÙØP [ ���?Â�� Ç �87 �/�Um�i Ø H+N»� � D�@~�g(ikjB�çô�#7�ç�4 a� �B-����P�4���" "!v�zl ½�#���J��� � Ø � N³�Ô�"�ýg(ikj�{Yoq��mnwqs"Úµ �Û��Üv�zl¢rkjypkwqmnjyrÿoql#>��$�$�'�� a�����$�1�������çô��Fl�jbj�L�m�iVu�jbs��� S���PO1);<�KA,Y��Mg(iVjbwqs�jyu K@ot�V�Nxz�Ujyl/oqp"rDm�iVj?=Vslnmwqs�rkjbs/ot�V�"s�wy¦k�zu�otm��Uw~p�¹ Ü �"¢�*¤~�h{Y}VxY="xzxzl%m�ikj��zlnwqmns�wq�"�z�MrV���-}Vl��zw~p@jy X}Votm��Uw~pÁÁ ¢ ¹ Ü �ÆÉ&媹 Ü � �/�Um�i É�� P [|��¾NÝ� Ø � � � K�Cf�g(ikj��boqlnj Ø ��C���wqs�s�jyln�.w~pVrVlhmnw�soqpVrkw~u��hoqxU¨_���Um�ikw~}km(rV�zs�jy��m��Uw~pVoqx#�.jbs�l��zl�mnjypV��j�¾��Ü¿��Cf�hoqpVrvm�iVj ��wqs�s�jyln�.w~pVr"�zpk|�rV���-}Vl��Uw~pv��w~pVlnm�oqp�m���l�É��ÞÝ éß µ �Õ-Ö�L��X×BEYG ý e ¬ j@oql�l�}Vu�j�m�iVotm�oqp �zpVr"�U�X��rV}Voqx�oq��m��U�qjyxz�;�ikw�w~l�jyl�rV�Us�jy��m��zw~pVl}k���hots�r��ikjyu��z�boqx�|qs�oqrV�Ujyp�m�l �Y�.w~l��Um��U�qjMm�oc¦k�zl��

Å [ �����*� Ç ��� �n� ù° � ù �Ø£ Ø ��� ������Â/Ã!� �n�$� � K ù �§Woql�l��zpk|�mnw�m�iVj xz�zu��zm%wq{9l�u�oqx�x`£ xUjyoqrVl(mnw@o�§�©¢ª���m¤�X�.j�u�wXrVjyx

ÁÁ ¢ ¹ Ü �ÆÃXà$É2ê¹ Ü ��¹ Ü � ��� �6Ã!��á � K�D~��/�Um�iÏÉ�� Ý éß µ oqpVr � ��� �n� Ý éµ Ø ��� ��Õ-Ö�L��X×BEYG � i1â LVd a G�ZtCYL j e�®kwqs��"oq��mnjbs��zoqxh�ikjyu�wqm�oc¦k�zl¢m�ikj_�qjyxzwX�b�Um�� rV�zl�mns�����"}km��Uw~p��zl/��w~pVlnm�oqp�mSÅ¢�����*� Ç �n� Kã oqpVr�m�ikjMm�}kspV�zpk|�s�otmnj ��pV��s�jyoqlnjyl��U{³�zp"rV�U�J�zrV}Voqxzlu�w��qj rVw��/pvm�ikj�|qs�oqr"�Ujypfm�oqpVr��zmhrVjy��s�jyoqlnjyl(�U{Pm�ikjb��u�w��qj�}k�X�5otsrVl��F��ikjyu�wq¨X�zpVj��l��zl� | [ ��� �n�y| Ü � ù ��£ Ú ��� ������Â/Ã!� �n�$� � K�K~�

ù <

Page 16: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

g(ikjyp�{Ywqs�£ªí C�o�§�©¢ª��¤u�wJrkjyx.{Hw~x�xUw��/lÁÁ ¢ ¹ Ü �ÆÃXà$É2ê¹ Ü � � �����ȹ Ü Ã!��á � KA+���/�Um�i�Ék� Ý éß µ oqpVr � ��� ��� Ý éµ Ú ��� ���g(iV�zl_j�¦koqu��"xUj rV�Us�jy��m�xU� ot�V�"x��Ujyl@mnwÿm�iVjj�¦X�.jbs��zu�jypfm�l�wq{h®kwqs�rQ�4 ����PO@�/�Um�iB�SOÀ�����þ�B�"oq��mnjbs���oS�*) ù :�,Y���®V}ks�m�iVjbs rkjbm�oq�zxzlMots�j|~�U�qjyp,�zpB);<A+R,��Õ-Ö�L��X×BEYG � i ] � ²-G ^ L j e�®kwqs(u�wJrkjyxz��pk| ��iVjyu�wqm�oc¦J��l��zp@oqu�wXjb�"otj��5j���w~u���"�zpkj M ¦koqu��"xUjylnD�oqp"r�KX�f�´�ìjq�U�~�5j��"}km�Å ¦ �����*� Ç �n�yÅ [ ���%�*� Ç ��R| ¦ ��� �n�y| [ ��� �� g/ikjl��boqxz�zpk|�xzjyoqrVl�mnw@oqrVr"�Um��Uw~pVoqx#j��.jy��m�l��zpvm�iVj���ikjyu�wqm�oc¦k�zl�mnjbs�u

� ��� �m� P [N � Ø ��� �w� Ú ��� �n�$� � K�<~�g(iV�zl��boqlnj��zl(oqxzlnw���w\�qjbs�jyrD����m�ikj�s�jyl�}VxUm�lhwq{9§Botm�xzot¨B);<RGI,PoqpVr�wq{ ¥ xUm¥) ù ,��Õ-Ö�L��X×BEYG"Í i I³²NIåäC ` ² a Z\²�×WCYd3OWC¡æTç ` CY²NI j e ¬ j�oql�l�}Vu�j�m�i"otm�o�lnmns�jyoqu&wq{jyxUw~pk|~otmnjyr_�"oq��mnjbs��zo�l�}V��i_oql%u���¦Xwq�"oq��mnjbs�zo �zl wqs��Ujyp�mnjyr_�zp�m�ikj/rV�Us�jy��m��Uw~pÊèϲ��H´ µ �gPw@rVjyl���s��U�.jMoqxz�U|~pVu�jypfm�mnw\�5ots�r,m�iV�zl�lnmns�jyoqu��5j ��ikwXw~lnj�m�iVj�m�}ks�pV��pk|�¨qjbspkjyx

Å3é���<P����Â�èk�����&Ç>Â�èV�$� Ù è Ù � ù ���{9m�ikj¢oq��m�}VoqxWrV�Us�jy��m��Uw~p5� Ç iVoql�o���w~u��.w~pkjypfm��zp�rV�zs�jy��m��Uw~pbè �Ywqs��¤èV��#m�ikjyp�m�iV��l��w~u��.w~pkjypfm��zl�jypVi"oqpV��jyr,oqpVr,m�ikj �Nots�m��z�bxUjMwqs��zjypfm�l��Um�lnjyxU{9�zp�r"�Us�jy��m��Uw~paèS�Ywqs��¤è#�s�jyln�.jy��m��U�qjyxU�V�� �«{�<,�zl(l�u�oqx�x.jypkw~}k|~i�m�ikjyp�m�ikj�rV���-}Vl��Uw~pvxz��u��Umh�zl

ÁÁ ¢ ¹ Ü �NÃ�Â4��ÿ¹ Ü�/�Um�i�pkw~pJ�¤��lnwqmns�wq�"�z�Mr"���.}"l��Uw~p

����¤�ç¢"�n� P [|3N9êë gª� ° P [N <kè�è @ g¥� ° P [N <kè�è E � K6ìí �g(ikj�rV���-}Vl��U�J�Um¤���zp¡m�ikj�rV�Us�jy��m��Uw~p è­wqs~�¤èR�zl@jyp"iVoqpV��jyr-�(��ikjbs�jyoqlv�Um_i"oql_m�iVjlnm�oqpVrVots�r��coqxz}VjuP [ � ��|3N³�h�zpvm�iVjMwqs�m�ikwq|qw~pVoqx³rV�Us�jy��m��Uw~p��Õ-Ö�L��X×BEYG�î i I³²NIåäEH²-dfL"EÆï"ZcLVOWCYGJI a j e ¥ pkw~p�xzwX�boqx.|qs�oqr"�Ujypfmb�"�/iV�z�i�u��U|~i�m�#j�u�jyoql�}ks�jyr��X�voqu�w�jb�No�oqxUw~pk|�m�ikjy�zs/l�}ks�{Foq��jq�V�boqp��.j�u�wJrkjyxUjyrv�X�¥Ã �����p�*�n�n� N° ÜMð ÄZñ�ò;ó µkô �(����� ð ô �*���¯É ô � � K�:~��/ikjbs�j ð �FC@��l�m�ikj�� üª�ç�/ "�'-��¥������#����'�2[Ï*������(&�Î�h��{ ð í C�m�ikjyp�m�i"�zl�j�¦J�Vs�jyl�l��zw~pot�V�Vs�w�¦J�zu@otmnjyl�m�ikj�xUwJ�boqxV|qs�oqrV�zjypfmÀÃ!�5�9®kwqs���ikjyu�w~lnjypVl��Um��z�qj�u�w\�qjyu�jypfmh�5j/mns�jyotm

ù :

Page 17: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

m�ikj�pkw~pJ�¤xUwJ�boqxP|qsoqrV�Ujyp�m ¥Ã �¡�zpDj�¦koq��m�xU�,m�ikj�l�oqu�j¢�hoy�;oql�Ã!�¡�zp,m�iVj¢�"s�jb�X�zw~}VlM ¦koqu��"xUjyl�DX� K�oqpVrÏ+k�Õ-Ö�L��X×BEYG"õ i OBCYZ\GXd a CY²#IPL"E�OWG�ZtC¡öJL a C¡öNG j e �hoq��mnjbs��zoJ��{Ywqs�j�¦Joqu��"xUjq�-ots�j�mnwXwl�u�oqxzxMmnw¡u�jyoql�}ks�jý��ikjyu����boqxM|qs�oqrV�Ujyp�m�l,oqxzw~pk|ÿm�ikjy�Us��.wJrk�¡oc¦k�zlb��g(ikjb�£u�w��qjm�iks�w~}k|~i�oMl��z|~pVoqx2=Vjyxzr¢oqpVr�m�ikjb��u�jyoql�}ks�jhm�ikj�l��z|~pVoqxX�cots���otm��Uw~p�oqxUw~pk|�m�ikjy�UsB�"otm�i��ü jypV��j�m�ikjMm�}ks�p"�zpk|�s�otmnj l�iVw~}Vxzrvrkjb�.jypVrvw~p�m�ikj�rV�Us�jy��m��Uw~p"oqx#rVjbs��U�totm��U�qj��

Á�÷ �3�þ�ø� � �3�¥ÂAÃ!����«pDm�ikj �Nots�ot�.w~xz�z��l��boqx��zpk|�m�iV��l/xzo��'xUjyoqr"l(mnw Á�÷ �.�u£ [ � ª �.£4��ÂIÃÊ«��5��g(ikj m���u�jrkjbs��U�totm��U�qj���l�wq{�xUw\�%jbshwqs�rVjbs5��w~u��Nots�jyr�mnw�m�ikj�|qs�oqrV�Ujyp�m5mnjbs�u�� g³w¢xUjyoqr"�zpk|�wqs�rVjbs�%j�wq�"m�oq�zp�m�ikj l�oqu�j�xz�zu��zmhoql/�zp M ¦koqu��"xUj¥KX�ù *£8��6 ��$&%��!âñ ���/�.4��- 6 B!-8���)

L�pkj���w~u�u�w~p�ot�"�Vs�w~oq��i@�zp�}VpVrkjbs�l�m�oqpVrV�zpk|�m�ikj(rk�JpVoqu��z�%�Vs�wq�.jbs�m��zjyl³wq{#s�jyoq��m��Uw~pk�mns�oqpVln�.wqs�m�jy �}Votm��zw~pVl�oqpVr,wq{%�%w~xzmn�yu�oqpVp;jy �}"otm��Uw~pVl�ots�j]���������% ��1�� �����A��);< ù ,���%��u�}"xUm��U�"xz���botm��Uw~p�wq{5��+����/�zm�i@�.w��5jbs�l5wq{e�@oqp"r���pfmnjb|qs�otm��zw~p-��w~pVj��boqp�rVjbs��U�qj�oqp�zp>="pV�zmnj�l�jy �}kjyp"��j�wq{�jy �}"otm��Uw~pVlM{Ywqs m�ikj1�~�¤u�w~u�jyp�m�l¢wq{�¹P�@��p m�ikj_jy �}Votm��zw~p {Hwqsm�ikj!N��fm�i3u�w~u�jyp�m m�ikj��¾N5� ù ����lnm¢u�w~u�jypfm¢ot�V�.jyots�lb��g³wD�bxUw~lnj_m�ikj@jy X}Votm��Uw~p"l{Hwqs%m�ikj�=Vslnm[N,u�w~u�jyp�m�lb�Xw~pkj/pkjbjyr"l�oqp@ot�"�Vs�wy¦k�zu�otm��zw~p�wq{.m�ikj¢�¾N�� ù ����u�w~u�jypfmb�g(iV�zlØën�bxUw~l�}Vs�j �Vs�wq�"xzjyu�ì��zlv�%jyx�x�¨Jpkw\�/p oqpVr£�/��rkjyxU� rV��l��b}Vl�lnjyr£�zpRmns�oqp"ln�.wqs�mm�ikjbwqs��q��þ,w~lnm oq}Vm�ikwqs�l�}Vl�j5���5�2�4�¢ots�|~}Vu�jyp�m�l�wqs s�jb|~}Vxzots�j�¦J�"oqpVl��Uw~pVl�mnw,�bxUw~l�jm�ikj,u�w~u�jyp�m@ln�Jlnmnjyu �Flnjbj�jq�ì|k�v);D�C�,/wqsº) ù C�,Y�� ü jbs�j,�%j��"s�jylnjyp�m�o m�ikjbwqs��3{Hwqs�bxUw~l��zpk|�m�iVj�u�w~u�jyp�m(jy �}"otm��Uw~pVl��/i"�z��i���lh�"oqlnjyr�w~p�o@u��zpV�zu@�U�yotm��Uw~p��Vs��zp"�b�U�"xUjq�®kwqs@�5w~xUmn�yu�oqp"pÿjy �}"otm��Uw~pVl�m�ikj��bxUw~l�}Vs�jv�Vs�wq�"xUjyu iVoql��.jbjyp�mns�jyotmnjyr­��pým�iVj

m�ikjbwqs��Rwq{ M ¦XmnjypVrVjyr g(ikjbs�u�wJrk�JpVoqu��z�bl��Flnjbj jq�ì|k��þ�ú}Vxzxzjbs�oqpVrÆ´�}k|q|qjbs���);< ù ,Y��¥ pRjyp�mns�wq��� {Y}Vp"��m��Uw~pVoqx��zl@u�oc¦k�zu��z�bjyr¡}Vp"rkjbs@m�ikj���w~pVlnmnsoq�zpfm�wq{E=k¦Jjyry=Vs�lnmÇNu�w~u�jyp�m�lb�_�«m �zl oql�l�}Vu�jyrSm�iVotm m�ikj;�¾NÝ� ù ����lnm¢u�w~u�jyp�m wq{hm�ikj@u@�zpV�zu��z�bjbs�ot�V��Vs�wy¦k�zu�otmnjylBm�iVjM�¾NÀ� ù ����lnm u�w~u�jypfmBwq{"m�iVj%mns�}Vjhl�w~xz}km��Uw~p-�Wg(iV�zlP|~�U�qjylBm�ikj�rkjyl��zs�jyr�bxUw~l�}ks�jq�9�«m�ot�V�.jyots�l5m�iVotmhm�ikjbwqs��Ujyl5{Ywqs(o�xzots�|qjMp�}"u �.jbs5wq{³u�w~u�jyp�m�l�ots�jM�bot�"ot�"xzjwq{Bot�"�Vs�wy¦k�zu�otm���pk|�lnmnjbjb�,|qs�oqr"�Ujypfm�l/oqpVr�l�ikwJ��¨Jl¥);:�:A,��®kwqs�m�ikj�mns�oqpVl��#wqs�m(jy �}"otm��Uw~pVl�{Ywqs/�"�zw~xUwq|~�z�boqx-�Vs�wq�"xUjyu�lhm�ikj¢pkjb|~otm��U�qjÎ` [ �8¶��¤�

pkwqs�u��boqp��.j�lnjbjyp�oql%oqp�jypfmns�wq����oql rkj�=Npkjyr��zp�m�ikjbs�u�wXrk�JpVoqu����blh�F�zm l�otm��zl*=Vjyl9oqpü �¤g(iVjbwqs�jyu��qjy X}Votm��Uw~p�� K�@~�n�� ¬ j��bxUw~lnjhm�ikjhu�w~u�jyp�m9ln�Xl�mnjyu ��� u@oc¦J�zu@�U�y�zpk|(m�iVjpkjb|~otm��U�qj^` [ �¤pkwqs�u �F�´�ìjq�Wu@�zpV�zu��z�y�zpk|�m�iVjT` [ �¤pkwqsu_�.}VpVrVjbs�m�ikj���w~pVl�mns�oq�zp�mPwq{=k¦Jjyr=Vs�lnmÆN.�¤u�w~u�jyp�m�lb��g(i"�zl%u@�zpV�zu��z�yotm��Uw~p��"otmnmnjyp"lhw~l��b�zxzxzotm��zw~pVlb�JiV�U|~i�{Hs�jy �}kjyp"�b�Ujylh�zpln�"oq��j�oqp"r�m��zu�j��/��xzxX�#j(l�u�w�wqm�iVjyr�w~}km�oqp"r�m�ikj(|~xUwq�NoqxVlnmns�}V��m�}Vs�j�wq{.m�ikj/lnw~xz}Vm��Uw~p�zl�jyu��"iVoql��U�bjyr-� ¬ j@�Vs�jylnjypfmMm�i"�zl��Vs�wJ��jyrV}ks�j�ikjbs�j�mnw��bxUw~lnj�m�ikj�l��Xlnmnjyu&{Hwqs�m�iVj=Vs�lnm³m¤�5w�u�w~u�jypfm�lh�Ymnwqm�oqxJ�.wq�"}Vxzotm��Uw~pMrVjypVl��Um���oqpVr��.wq�"}Vx�otm��Uw~p��"}J¦"��³g/ikj��bxUw~lnjyr

ù G

Page 18: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

ln�Xl�mnjyu �zl�o&û»�� 8�����ç�Ï�$!A�� a���¢�.�/i"�z��i���l(�5jyxzx�¨Jpkw���p,�zpDikjyotm/mns�oqp"ln�.wqs�m(m�ikjbwqs��q��«p¡);K�DA,��%j;|qjypkjbs�oqx��U�bjyrRm�iV�zl�ot�V�Vs�w~oq�i¡mnwÿ�bxUw~l�jDm�ikj�u�w~u�jyp�m�ln�Jlnmnjyu7otmvoqp��wqs�rkjbsy� ¬ j��/��xzx-l�}Vu�u@ots��U�bj�l�w~u�j s�jyl�}VxUm�l/��p,o�xzotmnjbs�l�jy��m��Uw~p-��®B�zp"oqxzxU���%j�ot�V�"xU�m�iV�zl(u�jbm�iVwXrvmnw_m�ikjMmns�oqpVln�.wqs�m�jy X}Votm��Uw~p�{Hwqs/�ikjyu�w~lnjyp"l��Um��U�qj u�w��qjyu�jyp�mb�

ãn_a^ V Ò tÔlmtÔoytýü>h�i�tacwl�%Td�talJÖ>t8«Jgaf¬ j_rkjyu�w~p"lnmns�otmnj�m�ikj_u�w~u�jyp�m �bxUw~l�}ks�j�u�jbm�ikwJr w~p m�ikj�j�¦koqu��"xzj�wq{ho��qjyxUwX�b�zm¤�¾n}Vu����Vs�wJ��jyl�lB�/�Um�i�=k¦Jjyr�ln�.jbjyr-�q�"}VmW�tots��zot�NxUj�rV�Us�jy��m��Uw~p��8U��ç��$����� � ��� V�);<�\�,Y��B��pm�iV�zl/�boql�j�¶õ��PO� µ � K �/�zm�ioP��QC_oqpVr ° � Ù ¶ Ù �øP µ � K ° Ü �N�/ikjbs�j ° Ü � Ù � µ � K Ù �g(ikj�m�}ksp­oqpk|~xUj;rV�zlnmns��z�"}km��Uw~p3�zl@oql�l�}Vu�jyrÿmnw3�.j���w~pVlnm�oqp�m1Å����%�*� Ç �]� Kã � ¥ l�Vs�jylnjyp�mnjyr �zpN);K�D�,��5m�iVj,u�jbm�ikwJr­rkjb�qjyxUwq�.jyrÿikjbs�j,�boqp­�.j�|qjypkjbs�oqx��U�bjyrÿmnw u�wqs�j|qjypkjbs�oqx�¨qjbspkjyxzl�Å'oqp"r�u�wqs�j�|qjypkjbsoqx-�qjyxUwJ�b�Um¤��lnjbm�l¿¶��g(ikj��zp"�Um��zoqx#�coqx�}kjM�Vs�wq�"xzjyu�{Hwqs(m�iVj�xz�zpkjyots�mns�oqp"ln�.wqs�m�jy �}"otm��Uw~p�s�jyoqrVl

¹ � ���?Â/ÿ¹ � | @ Û Ü° �]¹ E � � KRGq�¹B�"C�*�w�*�J�½� Q Ü ���w�*�k�$� � K�\~�

�«p�);K ù ,P�5j��Vs�w\�qjyr�m�ikj�{Yw~xzxUw\�/�zpk|Êþ,�¤g(ikjbwqs�jyuÏ�>��³G�²NZ\G�� � ÉÉ�� v ¹W�����*�w��� � v [[ � Á à 5 ÄRß�� à ¹B���$�*�p��� � [ ÉR� 7 sèC� � K�@~�g(ikj_�qjyxUwX�b�zm¤���¤u�w~u�jyp�m�l�wq{¯¹ÿots�j�rkj�="pkjyr ���~Ûbÿ#�B�/ikjbs�j1Û Ü �zl�rkj�="pkjyr �X�

� :~��oqpVr�m�iVj�iV�U|~ikjbs(u�w~u�jypfm�l/wq{�¹;ots�j�rkjypkwqmnjyr,���Û ê �����*�.� � Ä ß � ê ¹B���$�*�p�*�J�2ÉR��� ï� ù �������I�1N ��+RCf�Û ê à �����*�.� � Ä ß � ê � à ¹W�����*�w�*�k�2ÉR��� Ã������ ù �������/�1N»� ��+ ù �

� wqmnj�m�iVotmªÛ Ü �zl�l��boqxzotsy�rÛ ê �zl�o@�qjy��mnwqs�oqpVr5Û ê à��zl�o]D\��mnjypVlnwqsy� ¬ j�lnmns�jyl�l�m�iVj}Vl�}Voqx-l�}"u�u�otm��Uw~p���w~p��qjypfm��zw~p,w~p�s�jb�#jyotmnjyrD�zpVrV����jylb�gPwMrkjbs�U�qj�m�ikj5jy X}Votm��Uw~pVlW{Ywqs³m�ikj�=Vs�l�mPm¤�5wMu�w~u�jyp�m�l¯Û Ü oqpVr?Û ê �5j5��pfmnjb|qs�otmnj

� KRGq�hw��qjbsE¶�mnw@wq�Vm�oq�zpvm�iVj ��w~pVlnjbs��totm��Uw~p�x�oy�Û Ü� � Á à Û à ��C� ��+�D~�

þ,}Vxzm��U�"xz�z�botm��zw~p_wq{�� KRGq�h���Um�i]� ê oqpVr���pfmnjb|qs�otm��zw~p�|~�U�qjylÛ ê� � Á à Û ê à ���S|`Û ê � ��+�K~�

ù \

Page 19: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

gPw3�bxUw~lnj�m�iV��l�ln�Jlnmnjyu wq{Mm��%w3u�w~u�jyp�m@jy X}Votm��Uw~pVl,��+�D~�@oqpVr<��+�K~���5j,�hoqpfm_mnws�jb�"xzoq��jÀÛ ê à �þ¹.�� ¬ j(rkjbs��z�qjho�{F}VpV��m��Uw~p?}������J���$�*�p�*�J�³�/i"�z��i�u��zp"�zu��U�bjylPm�ikj¤` [ �8¶��pkwqs�u v }B���$�*�p��� � v [[ }"pVrkjbs m�ikj���w~pVlnmns�oq�zp�m m�iVotmJ}�������iVoql m�iVj�l�oqu�j�=Vs�lnm�u�w~u�jypfm�lÛ Ü oqpVrºÛ ê oqlJ¹�iVoqly�ªL�pV��j¢�%j�iVo��qj¢l�}"��i;o@{F}VpV��m��Uw~p5}��������5j�s�jb�Nxzoq��j?Û ê à �þ¹-����]Û ê à ��}������q��¬ j/�zp�mns�wJrV}V��j^]Pot|qs�oqpk|~��oqp¢u¢}VxUm��z�"xz�Ujbs�l� Ü ²v�H´ oqpVr� ê ²��H´�{Ywqs[Ã�� ù �������I�1NoqpVr�u��zp"�zu��U�bjþ3��}-���þ� ùD Ä�ß.} [ ÉR����� Ü 5 Ä�ß�}�ÉR���~ÛÝÜ 7 ��� ê 5 ÄRß�� ê }�ÉR���~Û ê 7 �

¬ j wq�Vm�oq�zp,oqp�j�¦X�Nxz�z�b�Umhs�jb�Vs�jylnjypfm�otm��zw~p�wq{Bm�ikj�u��zpV��u��U�bjbs��FlnjbjT);K ù ,³{Hwqs/rVjbm�oq�zxzl�}������������*�w�*�k�m� ù° 5 ÛÝÜ~�����*�.�w� NP [ ��� ê Û ê �����*�.�n� 7 � ��+�+��

� G�� L"Z� eû ? ��m/m�}ks�pVl�w~}Vm/m�iVotm�}��������zl�m�ikj��Vs�wA¾njy��m��Uw~p�wq{�¹Dw~p�mnw�m�ikj xz�zpVjyots/l�}k�"l��"oq��j| ù �*� K �������I�*� µ } ·,` [ �8¶����ý ? �«{³�5j u���pV�zu��U�bj�m�ikjM{F}VpV��m��Uw~p"oqxþ Ú ��}.���þ� ùD Ä ß ��}]� Ø � [ É������ Ü 5 Ä ß }�ÉR���~Û Ü 7 ��� ê 5 Ä ß � ê }�ÉR���~Û ê 7 �{Hwqs lnw~u�jhots��"�Umns�ots�� Ø ²��H´­�/�Um�i�m�ikj(l�oqu�j���w~pVlnmnsoq�zpfm�l9oql9ot�.w\�qj��%j(ots�s�U�qj5otm9m�iVjl�oqu�j�u@�zpV�zu��z�bjbs/��+�+����®kwqs�=k¦Xjyr Ø ²��H´­m�ikj�pkwqs�u v }B���$�*�p��� �>� Ø v [ �zl%o�u�jyoql�}ks�jwq{Wm�iVjMw~l��b�zxzxzotm��zw~p�ots�w~}VpVr�m�iVj�xUjb�qjyx Ø � ü jyp"��jq�7}������¢u��zpV�zu@�U�bjyl5w~l��b�zxzxzotm��zw~pVlb�� ? g/ikj�j�¦Jmns�jyu�}"u }������¢�zl(�zp"rkjbjyr�o@u���pV�zu¢}Vu��Jl��zpV��jMm�ikj�lnjy��w~pVr��tots��zotm��zw~pwq{�þ �zl�� [ þ���}.�n� ù �FCJ�gPw�rkjbs��z�qj�m�ikjDu�w~u�jypfmv�bxUw~l�}Vs�j��5j;��w~pVl��zrkjbs�m�ikj;lnjy��w~pVrRu�w~u�jyp�m_wq{Mm�iVj

u��zpV�zu@�U�bjbsÀ} ����� � Û ê à ��}������t�n� P [N ÛÝÜ�g7� ��+�<~�� w\� �5j_�bxUw~l�j�m�ikj_l��Xlnmnjyu wq{hm�ikj1=VslnmMm¤�5w;u�w~u�jypfm�l���+�D~��/��+�K~�M�X� oql�l�}Vu@�zpk|m�iVotm�Û ê à ��}-���NÛ ê à �þ¹-��/g(ikjyp-�Ns�jb�"xzoq�b��pk|�Û ê à ��pý��+�K~�(mnwq|qjbm�ikjbsM�/�Um�i ��+�D~�/|~�U�qjylo�xz�zpkjyots�¯�otmnm�oqpkjbw_ln�Jlnmnjyu Ï Ü� � Á à Ïyà � C�Ï ê� � Ý éµ Á ê Ï Ü � �S|�Ï ê � ��+�:~��/�Um�i��zpV�zm��zoqx.��w~pVrV�Um��Uw~p"lÏ Ü �"C��� �J�yÛ Ü �"C��� �$� Ï ê �"C��� �m�yÛ ê �"C��� �$� ��+&Gq�

ù @

Page 20: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

¬ j��zp�mns�wJrV}V��j@�bot�N�Um�oqx%xzjbmnmnjbs�l mnw;rV�zlnm���pk|~}V�zl�i;�.jbm¤�5jbjyp3m�ikj_u�w~u�jypfm�lv��Û Ü �*Û ê �wq{`¹�oqp"r,m�ikj�lnw~xz}Vm��Uw~pVl��»Ï Ü � Ï ê ��wq{ m�ikj�¯hotmnm�oqpkjbw�ln�Jlnmnjyu ��+�:~��¥L�{ ��w~}ks�lnjq�.�U{Û ê à���}.�(#�ÆÛ ê à~�þ¹.��m�ikjypý�»Ï Ü � Ï ê �(#� ��Û Ü �*Û ê ���g(ikj�jbs�s�wqs/�/i"�z��i�wJ�b�b}ks�l��zp�m�iV��lot�V�Vs�w�¦J�zu@otm��Uw~p��boqp��.j���w~p�mns�w~xzxUjyr-� ®Vwqs/m�iVotm(�N}ks��.w~lnj��%j�rkj�="pkj

� �þ��Û Ü �XÏ Ü oqpVr ' ê �þ�yÛ ê �Ï ê �oqpVr�oqp�jypkjbs�|q�

� Ý � � �F'~���þ� ùD Ä�� � ò � [ � NP [ ' ê ' ê ÉR�w� ��+�\~��«p�);K ù ,Pm�ikjM{Yw~xzxUw����zpk|�jbs�s�wqs(jyl�m��zu�otmnj iVoql(�#jbjyp,l�ikw���p-�>��³G�²NZ\G�� Í

� Ý � � ���$��� �$�F'k������� �n�4s,N Ú [µ P [DA| v à � Û Ü v [� é ��� Ü�. ����� � � ò � � ��+�@~�� �� ��º���B�$#�#�6��#�$� �� a�T�����7�� 8���% Ú µ �èC7O¯hotmnm�oqpkjbwÝ);:A,-}Vlnjyr_ln�Xl�mnjyu�l5wq{-m�ikj�{Ywqs�u ��+�:~��mnw¢u�wJrkjyxNikjyotm%mnsoqpVln�.wqs�m����Um�i="pV�Umnj�ln�.jbjyr-�Mg(ikjyp¼Ï Ü �zl�m�ikj�mnjyu��.jbs�otm�}ks�j¢oqpVrÿ�»Ï K �������4� Ï µ �/m�ikj�ikjyotm¿�"}J¦

oqpVr m�ikjSsotm��UwFÉ¡� P [�� |3N���l�m�ikjSj��.jy��m��U�qjýr"���.}"l��Uw~p£��w~pVlnm�oqp�m3�Flnjbj! qw~lnjb�Ni-�§ s�jb�y�Uw~l��¯) +�K�,-wqs���}ks�m��zp��X§%�U�V¨J�zpº);D�D�,.{Ywqs�m�ikj��"i��Xl��z�boqx.�zpfmnjbs��Vs�jbm�otm��Uw~pVlhoqpVrB);D�K�,��wqs�);K�C�,�{Ywqs/�"�zw~xUwq|~�z�boqx-�zp�mnjbs��Vs�jbm�otm��zw~pN��g(ikj¢rkjbs��U�totm��Uw~p�wq{9m�ikj�¯hotmnm�oqpkjbwvu�wJrkjyx�{Ys�w~u o_u�w~u�jyp�m��bxzw~l�}ks�j ot�"�Vs�w~oq��i

|~�U�qjyl³o�pkjb�3}VpVrkjbs�l�m�oqpVrV�zpk|hwq{km�ikj%s�w~xUj%wq{km�ikjh¯hotmnm�oqpVjbw�ln�Jlnmnjyu'�zp��N�Uw~xUwq|~�z�boqx~ot�V��"xz�z�botm��Uw~p"lb�9g(ikjMs�jyxUjb�toqp�m(�"ots�oqu�jbmnjbsl/ots�jMs�jyx�otmnjyr�mnw�m�ikj���pVrV�U�J�zrV}VoqxNu�w��qjyu�jypfm�#jyi"oy�J�Uwqs(wq{Wm�ikj }VpVrVjbs�xU�J�zpk|�l��#jy�b�zjylb�

ãn_'¦ V b�¨mf%oyc`i�h� »tÔzÏÒÓc»jnf%gªqÌt iR¨��Bf%lmz�t i;¡bTcwlpi�dRcwgü �zx�xUjyp�oqpVr�§Boq�zpfmnjbs�);K�<�,.iVoy�qjMlnm�}"rV�Ujyr�o¢rV���-}Vl��Uw~p��Noqlnjyr_u�wXrkjyx"{Hwqsh��iVjyu�w~lnjypVl��U�m��U�qj�u�w\�qjyu�jyp�m/�/ikjbs�j otm/iV�U|~i��#wq�N}Vxzotm��Uw~p�rkjyp"l��Um��Ujylhm�ikj���ikjyu�wqm�oc¦k�zl��zlhm�}ks�pVjyrwt�¡oqpVr3�"}ks�j_rV���-}Vl��Uw~pSrVw~u��zpVotmnjylb��g(iV��l u�wJrkjyx��boqp3�.j_��w~p"lnmns�}V��mnjyr �Y{Ys�w~u omns�oqpVln�.wqs�m%jy �}Votm��zw~pN�9�J�zo�o���wqs�s�jyln�.w~pVrV�zpk|�¯�otmnm�oqpkjbw�ot�V�Vs�w�¦J�zu@otm��Uw~p-��ªXw~xz}km��zw~pVlj�¦J�zl�m�|~xUwq�"oqxzxz�;oqpVrýpkw�� �"xUw\���¤}k�SwJ�b�b}ks�lb� ¥ l¢oqpSj�¦koqu��"xzj���w~pVl��zrkjbs¢o�m�}ks�p"�zpk|s�otmnj�wq{Bm�ikj�{Ywqs�u

|������"� ÷ �����þ�y| Ü 5 ù � NP [$# ��ÛÝÜ�� � �����%� ÷ � 7 ��/ikjbs�j # ��Û Ü ���zl�o@rVjypVl��Um���rkjb�.jypVrkjyp�m/lnjypVl��zm��U�X�zm¤�q�/g(iVj�{Y}"pV��m��Uw~p # �zl�oql�l�}Vu�jyrmnw¢iVo��qj�o �bjbs�w�otmhlnw~u�j

½Û Ü �FC oqpVr # ��ÛD�À�èC {Hwqs�C�H.ÛAH ½Û Ü � ¬ �zm�i�m�}ks�p"�zpk|D�C

Page 21: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

¨qjbs�pkjyx�Å¢�����*� Ç �T�þ� ° � K |������"� ÷"& � � m�ikj�u�w~u�jypfm��bxzw~l�}ks�jv�Vs�wJ��jyrV}ks�j�xUjyoqr"l�mnwSoû»�� 8�������]��������� ����?��������A�����>�$�" "�'-�������-����1���% � �� ����&���>���� "!]�����% "*���Ï Ü� � Á à Ï à � CÏ ê� � Ý éµ Á ê Ï Ü � ��| Ü à ù � µÝ é # �»Ï Ü � � ��� �1� � á¤Ï ê � # �»Ï Ü � � ��� ��Ï Ü Á ê ���� <�Cf�g(iV�zl_u�wXrVjyx�iVoql_�.jbjyp¡}"lnjyr¡��pv) ù KA,�mnw�rkjyl���s��U�.j��Notmnmnjbs�pR{Hwqs�u@otm��Uw~p¡��p¡l�x��zu�ju�w~xzrVl%oqpVr@�zp��"oq��mnjbs��zoJ��þ,wqs�jbw��qjbs��Vo pX}Vu�jbs��z�boqx"l���ikjyu�j�iVoql%�.jbjyp@rkjb�qjyxUwq�.jyr@mnwlnw~xU�qj_� <�Cf��ãn_'Ñ ��taäw¨nf%d�'Ýd�jnf%dÌÒÓcpoQf�lwiBb�g8cwz�®mdRfg(ikj_i"�U|~ikjbs¢wqs�rkjbs�u�w~u�jyp�m¢�bxUw~l�}ks�j@s�jy �}V�zs�jyl�s��z|qwqs�w~}Vl��#wXwq¨X¨qjbjb�"�zpk|Dwq{/oqxzx m�iVjs�jyxUjb�toqpfm�mnjypVlnwqs���pVrV�z��jyl��/iV�z�i��boqpVpkwqm��.j¢�zp"�bxz}VrkjyrDikjbs�jq� ¬ j�s�jb{YjbsMmnwB);K�DA,9{Hwqsrkjbm�oq�zxzlb�g(ikjRþ,�¤g(ikjbwqs�jyu&�Fg(iVjbwqs�jyuÓ+��%wq{Pm�iVj��"s�jb�X�zw~}Vlhlnjy��m��Uw~p��boqpv�.j�|qjypkjbs�oqxz�z�bjyr

mnw3m�}Vs�pV�zpk|ý¨qjbs�pkjyxzlTÅ �/iV����iRl�otm��zln{Y�­m�ikj�|qjypkjbs�oqx�oql�l�}Vu��Vm��Uw~p"l;�'Å ù �ª� �'Å�+���rkj�="pkjyr ot�.w��qjq��g(ikjSiV�z|~ikjbs�wqs�rkjbs,u�w~u�jypfmD�bxUw~l�}ks�jS�boqp<�.jSrkjbs��U�qjyr �zp£m�iVj{Hs�oqu�jb�%wqs�¨�wq{J]³ot|qs�oqpk|~�zoqpDu�}"xUm��U�"xz�zjbs�lb� �«m(m�}ks�p"l(w~}km(m�i"otm�m�ikj�lnmnjyoqrk��lnm�otmnjyl�wq{m�ikj/m��%w�u�w~u�jyp�m%�bxUw~l�}ks�j���¯hotmnm�oqpVjbw¢ln�Jlnmnjyu_� oqpVr�wq{.m�ikj�m�iks�jbj�u�w~u�jyp�m%�bxUw~l�}Vs�jots�j�rVjbmnjbs�u��zpkjyr@����oqp_jyxzxz�z�Vm��z�hjy X}Votm��Uw~p��F�´�ìjq�U�klnmnjyoqrk�@lnm�otmnjyl5wq{Po���wqs�s�jyln�.w~pVr"�zpk|rV���-}Vl��Uw~pD�Vs�wq�"xzjyu_�� ¬ j���w~pA¾njy��m�}ks�j�m�iVotmMm�iV�zl��zl�m�ikj��boqlnj�{YwqsMoqxzxBiV�U|~ikjbs�wqs�rVjbsu�w~u�jyp�m/�bxUw~l�}ks�jylb�

( nðy �-����9��) ù , ¬ � ¥ xUmb�N�5��oqlnjyr�s�oqpVrkw~u'�hoqxU¨�u�wXrkjyxJ{Ywqs ��iVjyu�wqm�oc¦J��l oqpVr�s�jyxzotmnjyr�r"���.}"l��Uw~pot�V�Vs�wy¦k�zu�otm��Uw~p-�*)>O,+B�R ��2O ² �"�����ç[R!t��@2� ù +&G�� ù G�G�� ù @�\�CJ�

);DI, ¬ � ¥ xUmb�³ªJ�zpV|~}Vxzots �.jbs�m�}ks��"otm��Uw~p�wq{�rV�U�#jbs�jyp�m��zoqxN�zpfmnjb|qsoqx"jy X}Votm��Uw~pVl%rkjyl���s��z�k��zpk|��N�zoqlnjyrvs�oqpVrVw~u��5oqxz¨Xlb�*)>O�ç�����������&[2� � O-+��� ��2O���K�D�D2� ù <;��+ ù � ù @�\ ù �

);KI, � �W�5jyxzxUw~u�wk� È ���4 "(ç�/.?�� a�������è ���/+��� �������� "� �����(�w�>�ç���!��6�T ��>� ² ���� 10�W�����7�Ý�S94(>�R "�"���"� ¥ rk�coqp"��jyl��zp_þDotm�ikjyu�otm��z�bl({Hwqs ¥ �V�"xz�UjyrvªJ�b�UjypV��jyly� ¬ wqs�x�rªJ�b�Ujyp�m��P="�t�Nªk�zpk|~ot�.wqs�jq� ù @�@�<X�

) +�, ü � ¯��f�5jbs�|�oqp"r�ú¢� ¥ �X�%s�w��/p-��¯hikjyu�wqm�oc¦J�zl%�zp M l���ikjbs�����iV�zo ��w~xz�´�~oqpVoqxU�Jl��zlB�X�m�iks�jbj��¤r"�zu�jypVl��zw~pVoqx#mns�oq��¨J�zpk|k�2.?�� "(ç����D�K�@2� <�C�C?�2<�C�+k� ù @RG�DX�

);<I,�§W�P�h�zxUjbsy�Î]�wJ�boqx oqpVr�|~xUwq�Noqx lnw~xU�tot�"�zxz�zm¤��wq{hlnw~u�j��"otsot�#w~x��z��l��Xlnmnjyu@l�u�wJrJ�jyxzxz��pk|ý�ikjyu�wqm�oc¦k�zlb� � ��-������������'�3+B�� ��2O¥���4�"O������ � #�#���O���\k� D~�$�þG ù <;�GI+�KX�ù @�@�\X�

D ù

Page 22: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

);:I,�¯��³¯hotmnm�oqpVjbwk�_ªJ}Vxzxzo���w~pVrV}V�y�Uw~pkj�rkj��boqxUwqs�jq� � "�4�����»����������O4+B�R 8OS��¶m�'��O5 �7�'-�O,+B���&�����t�rK2� \�K;� ù C ù � ù @A+�\X�)HG/,�¯��-¯5jbs��b�U|~pVoqp"�´�%´��N�«x�xzpkjbsy�"oqp"rDþ �#§%}VxU�J�Us�jyp�m��´�Ê�w�>�6+B�� ��>������ "� �����G�w������!�6�87¥���þ(> a� �:9��������Î�5ªX�Vs�zpk|qjbsy� � jb�3; wqs�¨#� ù @�@A+k�);\I,�©�� ¯��.¯hikjyp-�>´��Ûþ �N®Vwqs�r-�"oqpVr,§³�ºg��#¯h}Vu�u@�zpk|~lb� þDotm�ikjyu�otm��z�boqx³u�wXrVjyxzlh{Hwqsu�wqm��zxzjM�"oq��mnjbs��zoqx-mnsoqpVln�.wqs�m��zp����Jxz�zpVrVs��z�boqxNm�}k�.jylb�mUÀç�Ô#���'�� F� ù @�@�\X�

);@I,�©�� ¯��J¯hikjyp-��´ �ÛþS�X®kwqs�r-�JoqpVr�§W�ºgM�J¯h}Vu�u���pk|~lb�N§Wjbs�m�}Vs��"otm��Uw~p�j�¦J�"oqpVl��Uw~p�wq{¥ xUm�öºlM��jyxzxB�"oqxzoqp"��j¢jy X}Votm��Uw~p"l�s�jyrV}V��jyl�mnwDªXjb|qjyx8öºl ù r�jy �}"otm��Uw~p�{Hwqs�l�iVoqxzxUw\���iVjyu�w~otmnmns�oq��m�oqp�m�|qsoqrV�Ujyp�m�lb�S� Ì � +<)>O � #R#��PO-+B�R ��2Oz��<�@k� ù �$� K�<;�2<RG�� ù @�@�@X�

) ù CA,�´���ú�oq}Vmns�oy� oqpVr= k� �ý] �G]P�Uw~pVly�>+B�� ��>������ "� ����� � ������!A�$�'�B�����?.�(�1����"�$���+��4 ��2���A�[����¿�����8�����$�E����� ���ç���������P�ç[R!c�"�qw~xz}Vu�j8@��Î�WªX�"s��zpk|qjbsy�X�5jbs�xz�zp-� � jb�; wqs�¨#� ù @�@�KX�) ùqù ,�´��`ú��z��¨X�zp"lnw~p-� ¥ |qjypkjbs�oqx��U�bjyr;mns�oqpVln�.wqs�m�u�wXrVjyxP{Ywqs��"�zoqlnjyr���jyxzxWu��U|qs�otm��zw~p

�zp�oqpDoqpV�zlnwqmns�wq�N�z��jypf�J�Us�w~pVu�jypfmb�*)>O,+B�� ��O ² �"���PO��7+RC� @RG�� ù K�<X�wD�C�C�CJ�) ù DI,�´��º���eú��z��¨X�zp"lnw~p oqpVr~´��ºg��.g³s�oqpV X}V�zx�xUwk��gPsoqpVln�.wqs�m�jy �}Votm��zw~pVlMoqpVr �zpVr"�z��jyl

{Ywqs�s�oqpVrkw~u oqpVr��N�zoqlnjyrÿ��jyxzx�u��z|qs�otm��Uw~p��"oqlnjyr�w~pÿl��zpk|~xUj���jyxzxh�Vs�wq�#jbs�m��Ujylb�� Ì � +A)O � #R#��PO-+B�R ��2Oz��<�<2� ù + ù @;� ù +�<A+k� ù @�@�<X�) ù KI,B;��nú�w~x�ot¨�oqp"rÿgM� ü ��xzxUjyp-�3¯�otmnm�oqpkjbw u�wJrkjyxzl�{Ywqs���ikjyu�wqm�oc¦J�zly�hpX}Vu�jbs�z�boqx

lnw~xz}Vm��Uw~p�oqpVr��"otmnmnjbsp�{Hwqsu�otm��Uw~p-�C)>O,+B�R ��2O ² �"����O��7+�:k� D~�$� ù <�K;� ù GACJ�pD�C�CRKX�) ù +�,¢§W�H´���®B�zl�ikjbsy�J´ ��þ�jbs�¨Xx´�5oqpVrQ�¢�À��jbs��zl��i-�ED�}Voqp�m��Um�otm��U�qj,oqp"oqxU�Xl��zl�wq{���jyx�x

u�wqm��zx��Um¤�¢oqpVr���ikjyu�wqm�oc¦k�zl%�zp�ú�����m¤�qw~lnmnjyxz��}Vu rV�zl���w~��rkjy}Vu'���¢}Vl��zpV|�oqp��zu@ot|qj�Vs�wJ��jyl�l���pk|�ln�Jlnmnjyu oqpVr¢o�pkw��qjyxk�ikjyu�wqm�oc¦k�zl9��i"oqu �.jbsB�Vs�w\�X�zr"�zpk|/lnm�otm��zw~pVots����iVjyu��z�boqx-|qs�oqr"�Ujypfm�ly�C)O¤ûn���'� ² � ���POz� ù CR\2� @RG�K;�2@�\A+k� ù @�\�@X�

) ù <I,�´��N®kwqs�r�oqpVr3�� À����O%i�mnmn�e�þ���c�b}ks��Ujq�º�ikjq�ì�J�Us�|~�zpV��oJ�ìjyrV}��c�bxUjb���c�bxzjb�"u�oq�zp-�ºi�m�u�x´�) ù :I,�´��Ûþ �.®kwqs�r;oqpVr5ú¢� ¥ ��]Poq}k�#jyp��"}ks�|qjbs���þ,jyoql�}ks�jyu�jypfm�wq{9�"oq��mnjbs���oqx�s�oqpVrVw~u

u�wqm��zx��Um¤��oqpVr���ikjyu�wqm�oc¦J�zl/��w�jb�@�b�zjypfm�l�������kot�V�Nxz�z�botm��Uw~p�wq{Bl��zpk|~xUj���jyxzx.�"oql�jyru�otm�ikjyu@otm��z�boqxPu�wJrkjyx´� ² � �� a�ç���������PO ² � �/���2[�O��%KRG&� :�: ù �2:RG�DX� ù @�@ ù �

) ù G/,�´��Ûþ �³®Vwqs�r-�B���H´ �P§�i"�zxzxz�U�Nlb�F V� ¥ ��D�}V�zp"p-�-oqpVr�ú¢� ¥ �¨]Poq}k�#jyp��"}ks�|qjbs��_þ�jyoc�l�}ks�jyu�jypfm¢wq{(�"oq��mnjbs�zoqx�soqpVrkw~u u�wqm��zxz�Um��SoqpVr3��ikjyu�wqm�oc¦k�zl���wXjb�@�b�Ujyp�m�l����Î�lnmnwq�V�.jyrJ�8�"w�� rV���-}Vl��zw~p��iVoqu��#jbs(oql�l�oy�q� ² � �� a������������O ² � �/���2[�O���KRG&� :A+&G��2:�:�CJ�ù @�@ ù �

) ù \I,¢gM�`��oqxzxzo���oqpVr�����´�oq}k|qjyx´��ªJ�boqx��zpk|��tots��zot�"xUjyl�oqpVr;lnm�ot�"�zxz�zm¤�vwq{ if�X�.jbs��.w~xz�z�{Ys�w~pfm�ly�S� Ì � +G)>O,+B�R ��2O � ����� �F�%K�D2� ù �2D�@X�eD�C�C�CJ�

D�D

Page 23: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

) ù @I,�ª#�»��w~xzrVlnmnjy��p-�ºL�p r"���.}"l��Uw~p �X�SrV�zl���w~p�m��zpX}kw~}Vl�u�w��qjyu�jyp�m�l�oqpVr m�ikj_mnjyxzj��|qs�ot�"i�jy �}Votm��zw~p-� � (7��� 8O�)O,+��ç���O � #�#���O+B�� ��2O��7+>� ù D�@;� ù <�:X� ù @�< ù �

);D�CA, ü �¯��s�oqr��ºL�pSm�ikj_¨X��pkjbm��z��m�ikjbwqs�� wq{�s�ots��P=VjyrS|~oqlnjylb� û»������OJU�(*� � #R#��PO+B�R ��2Oz��D2� K�K ù ��+RC�G�� ù @A+�@X�);D ù , k�Ûþ ����s�jbjypf�.jbs�|,oqp"r ¬ � ¥ xUmb�_ªJm�ot�"�zxz�Um��,s�jyl�}VxUm�l�{Ywqs�o�rV�U�.}Vl��Uw~p,jy X}Votm��Uw~p

�/�Um�i�{Y}Vp"��m��Uw~pVoqx-rks��z{Hm�ot�V�Vs�wy¦k�zu�otm��zpk|�o@��ikjyu�wqm�oc¦k�zl�u�wJrkjyx´�Î�76���>��O � +~�V�K�C�CV� ù �$� D�K�<;�2D�<�\X� ù @�\RG��);D�DI,�þ � M �&��}ks�m��zp�oqp"r ¥ � ¯��q§%�U�V¨J�zp-� ¥ |qjypkjbs�oqxJm�ikjbwqs���wq{#ikjyotm9��w~pVrV}V��m��Uw~p����Um�i="pV�zmnjM�5o��qj�ln�.jbjyr-� � *���2O¨���� "� �������,+������2O � ������O���K ù � ùqù K;� ù D�:X� ù @�:�\X�);D�KI,¢©��º§W� ü oqrkjyxUjbsy�n´/jyoq��m��Uw~p�mnjyxUjb|qs�ot�Ni�jy �}"otm��Uw~pVl�oqpVrvs�oqp"rkw~u��5oqxz¨�ln�Jlnmnjyu�ly�

��p_ª#�H k�q�toqp_ªXmns�Ujyp�oqpVr_ª#�Ûþ �I@9jbs�rV}V�Xp�]³}Vpkjyx´�qjyr"�Umnwqs�lb�7�� 8�4���2���� "� �E��������#>��W "� ���À�% "$(7�/ "(2ç���5�6�J7¥!��������"�$�����7!A�� a������n´(w\�~oqx ¥ �boqrkjyu �3wq{/m�iVj � jbm�ikjbs��xzoqpVr"lb� ù @�@�<X�

);DA+�,¢©��º§W� ü oqrVjyxUjbsy�w´/jyoq��m��Uw~pvmns�oqpVln�.wqs�m5ln�Jlnmnjyu�lb����pK@��"¯hot�"oql�lnw¢oqp"r]L¢�ú��Ujb¨��u�oqpVp�� jyrV�Umnwqs�ly�L+B�� �������� "�"���Ï�'�>�a#���*� �I©�!&©�� ������[�!t���"ot|qjylT@�<;� ù <�CJ�h¯5�nþ M]�jbm�}Vs�jyl ù @�@RG��.®Wxzwqs�jypV��jq�#ªX�"s��zpk|qjbsy� ù @�@�\X�

);D�<I,¢©��º§W� ü oqrkjyxUjbsy� � w~pVx��zpkjyotsh�Vs�wq�"ot|~otm��zw~p,�zpvs�jyoq��m��zw~p�mns�oqpVl��#wqs�mhl��Xlnmnjyu@lb� ��pª#��´/}Voqp;oqpVrÌ��� ¬ w~xU¨qw\�/�z���q�.jyrV�Umnwqslb�M7¥� üª��ç���� "�"���`�S94(>�R "�"���7� � �" �� � #�#��þ��W���R "�"���7�� 8� ² � ���P�ç[�!t�-�"ot|qjyl�D�< ù �2D�<RG��Pg(iVj�®B�UjyxzrVl/�«p"lnm��Um�}kmnjÎ]�jy��m�}ks�j�ªXjbs��zjylb�¥ þ;ª#� ù @�@�@X�

);D�:I,¢©��º§W� ü oqrkjyxUjbs��fg(ikj s�w~xUj wq{Vu��U|qsotm��Uw~p oqpVr���w~pfm�oq��m9rV��lnmns��U�"}Vm��Uw~p��zpMjb�"�zrVjyu��z�ln�Vs�jyoqr-�q�«p�¯��c¯hoql�m��zxzxUwt�«¯hi"oy�qjb�%oqp"r ü �ºgM�\�hoqpk¨Jlb�cjyrV�Umnwqs�lb� ² �"�����R �������� "� �����+B�;�������'�2[ � #R#��þ� ���� "� ���>���'�ON?���1���P������������($�" "!t�#ªX� ¥ þ ��D�C�CRKX�

);DRG/,¢©��º§W� ü oqrkjyxzjbshoqp"r@gM� ü ��xzxUjyp-�`´(jyoq��m��zw~p_mnsoqpVln�.wqs�m�jy X}Votm��Uw~p"l��/�zm�i@��w~}k�"xzjyrrk�JpVoqu��z�bl�oqp"r��/�Um�i�lnjyrVjypfm�ots���lnm�otmnjyly��D�C�CRKX�%��pv�Vs�jb�"ots�otm��zw~p-�

);D�\I,¢©��º§W� ü oqrkjyxUjbs oqp"r þS�¨]�jb�/�zly�_ªX�"otm��zoqx9rk�JpVoqu����bl�wq{5m�iVj�rV���-}Vl��U�qj�xUwq|~�zl�m��z�jy X}Votm��Uw~p��/�Um�i l�jyrkjypfm�ots�� ��w~u��"ots�m�u�jypfmb�Jû»���������"��� � #R#��POP+B�� ��2O � (7�� a�����!c�D�C�CRKX�5mnw@ot�V�.jyotsy�

);D�@I,¢§W� ü jyl�lMoqp"r ü � ¬ jy�zpf�.jbs�|qjbs���¯5w~pf�qjbs�|qjypV��j�mnw�ln�"otm��zoqx���mnjyu��#wqsoqxW�bx��zpkjyl��zp®B�zl�ikjbs/öºl9jy �}"otm��Uw~p��/�zm�i�m��zu�j����.jbs��UwJrV�z�J=Vm�pkjyl�lnjylb�Q)O�+��� ��2O ² � ���POz�>D�\2� \�K;�2@�\X�ù @�@�CJ�

);K�CA,¢gM� ü �zxzxUjyp-�BD�}"oqxz�Um�otm��U�qj�oqpVoqxU�Jl��zl�wq{%l�jyu��zxz�zpVjyots�¯hotmnm�oqpkjbwDln�Xl�mnjyu�lb�R+��� ��2O+B�;�����þ�S+��4 ��2O � #R#��PO¯����� O���\k� K~�$� <�C�G��2< ù @X� ù @�@�\X�D�K

Page 24: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

);K ù ,¢gM� ü ��xzxUjyp-�wL�pl` [ �¤�bxUw~l�}Vs�j�wq{Pmns�oqpVln�.wqs�m%jy �}Votm��zw~pVl��9g(ikjM¯hotmnm�oqpkjbw��bxUw~l�}Vs�jq�7¥�È����ç�� a������� û»���% 8O7¥!���O��7!A�� 8O»������ ��� ² �%D�C�CRKX�%oq�b��jb�Vmnjyr-�);K�DI,¢gM� ü �zxzxUjyp-��L�pC` [ �¤�bxUw~l�}Vs�j�wq{ mns�oqpVl��#wqs�m�jy X}Votm��Uw~p"l��ÿg(ikj;|qjypkjbs�oqx��boqlnjq�7¥�È����ç�� a������� û»���% 8O7¥!���O��7!A�� 8O»������ ��� ² �%D�C�CRKX�%oq�b��jb�Vmnjyr-�);K�KI,¢gM� ü �zxzxUjyp-�Sg³s�oqpVl��#wqs�m�jy X}Votm��Uw~pVl¢�/�zm�iýs�jylnm��zpk|��"iVoql�jylb�3�À(*�$#rO8)>O � #R#��PO+B�R ��2Oz��D�C�CRKX�%mnw@ot�"�#jyots��);KA+�,¢gM� ü �zxzxzjyp�oqp"r ü �þ����L�m�iVu�jbs���g(ikj�rV�U�.}Vl��Uw~pvxz�zu��Um5wq{Bmns�oqpVln�.wqs�m�jy X}Votm��Uw~p"l

rkjbs��z�qjyr�{Hs�w~u �qjyxUwJ�b�Um¤�ª¾�}Vu����Vs�wX��jyl�l�jylb�p� Ì � +T)>O � #R#��PO�+B�R ��2Oz�: ù � K~�$�þG�< ù �G�G�<X��D�C�C�CJ�);K�<I,¢gM� ü ��xzxUjyp;oqpVr�©��#§Boq�zp�mnjbsy�?��xzwq�"oqxWj�¦J�zl�mnjypV��j�{HwqsMo��"ots�ot�.w~xz�z� ��ikjyu�wqm�oc¦k�zl

u�wJrkjyx.�/�Um�i��Vs�jb�qjypfm��Uw~p�wq{³w\�qjbs���s�w���rV�zpk|k� � ��-AO � #R#��PO-+��� ��2O���D�:2� D�\�C?�2K�C ù �D�C�C ù �);K�:I,¢gM� ü ��xzxUjyp-�X¯��2´/w~iVrkjq�JoqpVr_®5��]P}km�l���ikjbsy� M ¦k�zlnmnjypV��j�wq{��5jyot¨@lnw~xz}km��zw~pVl {Ywqs�o

i����.jbs��.w~xz�z�Mu�wXrkjyxP{Hwqs���iVjyu�w~lnjypVl��zm��U�qj�u�w\�qjyu�jyp�mb�R)>O-+��� ��2O � �����PO � #R#��POz�D�:�C� ù G�K;� ù @�@X�pD�C�C ù �);KRG/,¢gM� ü �zx�xUjypMoqpVr¢¯��cªJ��i"u�jy�zlnjbsy�&�MxUwq�"oqx~j�¦k�zlnmnjyp"��j�{YwqsPm�ikj5�bxzoql�l��z�boqxf��ikjyu�wqm�oc¦k�zl

u�wJrkjyx.�/�Um�i]=NpV�Umnj�l�oqu��Nxz�zpk|¢s�oqrV�z}"lb�nD�C�CRKX�%�zpv�Vs�jb�"ots�otm��Uw~p-�);K�\I,¢gM� ü �zxzxzjyp3oqp"r ¥ ��ªXmnjb�qjypVlb� ü �X�.jbs��.w~xz�z��u�wJrkjyxzl�{Ywqs_��iVjyu�wqm�oc¦J��l@�zp ù �¤r��.?���7���'������ � ������!I���È��µ4�S�ç���VU5����É� � #R#��þ� ���� "� ���>�Î� ù � ù �$�;+RCR@;��+�K�KX�wD�C�C ù �);K�@I,�ú¢� ü wqs�lnm�u@oqpVp-� ®ks�w~u ù @RGAC }Vpfm���x��Vs�jylnjyp�m���g(ikj¡©�jyxzxUjbsn�«ªXjb|qjyx@u�wXrVjyx

�zp �ikjyu�wqm�oc¦k�zl<oqpVr �Um�l ��w~pVlnjy X}kjypV��jyl<�� )>�A��ç����©$����"���& a� ����W7S+WXP�ù CR<k� K~�$� ù CRK;� ù :�<X��D�C�CRKX�

) +RCA,�ú¢� ü wqs�lnm�u�oqp"p-�¡®ks�w~u ù @RGAC3}Vp�m��zxh�Vs�jylnjyp�m���g(ikj,©Mjyx�xUjbsn�«ªXjb|qjyx(u�wXrkjyx/�zp��iVjyu�wqm�oc¦J��l�oqpVrD�Um�l(��w~pVl�jy �}kjyp"��jyl/���Î��)>�A��ç����©���$� ���& a�!����Y7S+WXP�%D�C�CRKX�(�zp�Vs���pfmb�

) + ù , ü �H k� ü �5oqpV|k�h©���©�oqpk|k�hoqpVr ¥ �5ªXmnjb�qjypVly����xUwq�Noqxhj�¦J�zl�mnjypV��j�wq{M�bxzoql�l����boqxlnw~xz}Vm��Uw~pVl�{Hwqs�o�i����.jbs��NxUw~�z�,�ikjyu�wqm�oc¦k�zl�u�wJrkjyx�oqpVr£�Um�lv�"otsot�#w~x��z�,xz��u��Umb�+5U Ì U�*�Z#�$���% F�rKA+k��D�C�CRKX�

) +�DI,¢©��� �úwqs�|qjypVlb� ¥ p�oqln�Ju��Vmnwqm��z� j�¦X�"oqp"l��Uw~pM�zp�m�ikj�m�ikjbwqs���wq{XpVjy}kmns�w~pMmns�oqp"ln�.wqs�mb�û»������OpU�(2ç� � #�#���O+B�� ��2O�� ùqù � D ù @;�2DA+�DX� ù @�<�\X�) +�KI,�ú¢�Hú¢�F ~w~lnjb�"i�oqpVro] �"§�s�jb�y�Uw~l��´� ü jyotm��5o��qjylb���S��-I� � � ���6�Y+B������$�ºUÀ��!A�$� ���Î�: ù �;+ ù �G�KX� ù @�\�\X�

DA+

Page 25: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

) +�+�,�þ �t© oq�t� ¥ lnmnwX�iVoqlnm��z�hu�wJrkjyx�s�jyxzotmnjyr¢mnw�m�ikj5mnjyxUjb|qs�ot�"ikjbs[ZÛl³jy X}Votm��Uw~p����S�4��VA!+B��(2�� 8���'�\)>O,+B�� ��Oz�7+>�;+�@RG��2<�CR@X� ù @�<�:X�) +�<I, M �º®5�J©MjyxzxUjbshoqpVrÇ] � ¥ �kªXjb|qjyx����«p"�Um��zotm��Uw~p@wq{Pl�xz�zu�j/u�w~xzr_ot|q|qs�jb|~otm��Uw~p��J�Ujb�5jyr

oql/oqp,�zpVlnm�ot�"��xz�Um¤�q��)>O4�w��ç���O ² � ���P�ç[�!t��D�:2� K�@�@;��+ ù <X� ù @RGACJ�) +�:I,�þ � ¥ �t©Ms�oql�pVw~lnjyxzln¨J�z�´�RUS�A���� "��-��À������( "� ���>�S�6�S]p#>��*�� 8��J��9�(7�� "� ���>�Î�f§W� � w�wqs�rk�

ikwt�B�>��s�w~pV�zpk|qjyp���g(ikj � jbm�ikjbsxzoqpVrVlb� ù @�:A+k� M pV|~xz�zl�i�mns�oqp"l�xzotm��Uw~p@wq{e´/}Vl�l��zoqpwqs��U|~��pVoqx´�

) +&G/,�þ � ¥ ��]�jb�/��l oqpVrØ�¢�9ªk��iVu��zmn�q�v�h�Uw~xzwq|~�z�boqx%��pf�toql��Uw~p wq{�oqpSwqs�|~oqpV��l�u ���Um�ilnjb�"otsotmnj¢u�wq�N�zxUj oqpVr;lnm�otm��Uw~p"ots��,lnm�otmnjyl��/þ,wJrkjyxz�zpk|_oqpVrDoqp"oqxU�Xl��zlb�G¶e��$���t�ùqù � ù �2D�<X� ù @�@�:X�

) +�\I,�ª#�NþDoq�Ujbsn�¤§Boqot�.j�oqpVr,gM� ¬ oqpVpkjbsy��ªX�N�zpkwJrVoqx.rkjy��w~u��.w~l��Um��Uw~p�{Hwqs�m�ikj��boqiVpk�iV�zx�xzots�r�jy X}Votm��Uw~pR�zp­iV�U|~iVjbs_rV�zu�jypVl��Uw~pVly�W�Nots�m_� ��§ s�wq�"ot�"�zxz�zm¤��oqpVr­�hoy�qj��xUjypk|qm�i�jylnm��zu�otmnjq��û»������OQ+B�� ��2OpU��& ���Oz� ù @�<2�;+�K�<;��+�:A+k� ù @�@�\X�

) +�@I,¢§W� ¥ ��þ,ots�¨qw��J�z��i-��¯�� ¥ �S´��zpk|~ikwq{Yjbsy�MoqpVr1¯���ªJ��iVu�jy�zlnjbsy� �������"�$������(7�4 8���S94(7�� "�"���>��5ªJ�Vs��zpk|qjbs�� � jb�3; wqs�¨#� ù @�@�CJ�);<�CA,�ú¢�JþD�ziVoqx�oql%oqpVr_��� ¬ jy�U�.jyx���þD�ziVoqx�oqlb�¨¶e��(������ "�"���>�¥�6�¤�S����� �� "�"���ONE!���*����!�W������� ���Î�ÀL�¦J{Hwqsr � pV�z�qjbs�l��Um��_§ s�jyl�lb� ù @�\�KX�);< ù ,��Î� þ,ú}Vx�xUjbs oqpVr gM�B´/}V|q|qjbs��´� �S�� "�"���������nô� a�����&�F�Ù�w���$������!A�������"����

ªX�Vs�zpk|qjbsy� � jb���^; wqs�¨#��DtpVrvjyrV�zm��Uw~p-� ù @�@�\X�);<�DI, k�Hú¢�NþD}ks�soy�q�2+��� �������� "� ����� ² � ������[�!t�hªX�"s��zpk|qjbsy� ù @�\�@X�);<�KI, ü �þ�¢��L�m�iVu�jbs��Vª.�H´��>ú�}Vpf�Notsy�VoqpVr ¬ � ¥ xUmb�9þ�wJrkjyxzlhwq{WrV�zln�.jbs�l�oqxN�zp��"�zw~xUwq|t�

�z�boqx�l��Xlnmnjyu@lb�*)O+B�� ��2O ² � ���POz��D�:2� D�:�K;�2D�@�\X� ù @�\�\X�);<A+�, ü �þ�¢�rL�m�i"u�jbs/oqpVr�gM� ü �zx�xUjyp-�9g(ikj�rV�U�.}Vl��Uw~pvxz�zu��Um5wq{Bmns�oqpVln�.wqs�m�jy X}Votm��Uw~p"l

�F�z�H�$� ¯hikjyu�wqm�oc¦k�zl@jy �}Votm��zw~pVlb��� Ì � + )>O � #R#��POV+B�R ��2Oz��:�Dk��+��$� ùqù D�D;� ù D�<�CJ�D�C�CRDX�);<�<I, ü �þ�¢�wL�m�iVu�jbs�oqpVr ¥ ��ªXmnjb�qjypVlb� ¥ |q|qs�jb|~otm��Uw~p-�-�"xzw��/}k�;oqpVr;��w~xzxzot�"lnj��(g/ikj

¥ ��¯¿öºl�wq{¢m�oc¦J�zlD�zp£s�jy�zpV{Hwqs���jyr£soqpVrkw~u �hoqxU¨Jlb� � Ì � + )>O � #R#���O6+��� ��2O��<RG&� ù C�+�+O� ù CR\ ù � ù @�@RG��);<�:I,¢©���§Boq�zp�mnjbs oqpVrSg�� ü �zxzxzjyp-�_@9w~x�}Vu�j��8="xzx��zpk|_oqp"r  �}kwqs}Vu��¤lnjypVl��zpk|v�zp u�wJrJ�

jyxzl/{Hwqs���ikjyu�w~l�jypVl��Um��U�qj¢u�w��qjyu�jypfmb�Çû»��������� ��� � #R#��PO+��� ��2O � (>��� 8O���D�C�CRKX�oq�b��jb�Vmnjyr-�

D�<

Page 26: Nonlinear Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen1 and KP

);<RG/,�¯��Wª.��§Wotm�x�ot¨N�1´�oqpVrkw~u �hoqxU¨��/�Um�i��.jbs�l��zlnmnjypV��j�oqpVr j�¦Jmnjbs�pVoqx��"�zoqlb� ² (����O+B�R ��2O ² �"��#%��!A��O�� ù <2� K ùqù �2K�K�\X� ù @�<�KX�);<�\I,¢©��k§Wjyots�lnw~p-�9g(iVj��Vs�wq�"xUjyu wq{Pm�ikj�s�oqpVrVw~u �hoqxU¨#��.?�� "(ç����G�D2� D�@A+@oqpVrÝKA+�DX�

ù @�CR<X�);<�@I,�þ � ¥ �´/�U�qjbs�wk�2´ �ºgM��g³s�oqpV X}V�zxzxzwk� ü �Ûþ �X�h}kjbmnm�pVjbsy�Xoqp"rTú¢� ¥ ��]³oq}J�#jyp��"}ks�|qjbsy�

g³s�oqpVln�.wqs�m/u�wXrkjyx�l�{Hwqs���ikjyu�wqm�oq��m��z����jyxzx��.wq�"}Vx�otm��Uw~pVlh�"oqlnjyr,w~pD��pVrV�U�J�zrV}Voqx��jyxzx-�.jyiVo��X�zwqsy�uûp�����Ow���&[�O������ Oz�>+�+k� ù @�\�@X�

);:�CA,�þ �H k�Jªk��iVpV�zmn�bjbsy�Ng(iVjbwqs���wq{.��w~p�m��zpX}V}Vu soqpVrkw~u �hoqxU¨Jl oqpVr�ot�V�Nxz�z�botm��Uw~pVl mnw��iVjyu�wqm�oc¦J��lb�ÀUÀ��!A�$� ����������-AOp�h�>+�\2� D�<�<�K;�2D�<�:�\X� ù @�@�KX�

);: ù , ü �WªJ��i��5jbm�xz�z��¨#�_g³s�o��qjyxzxz�zpk|�{Ys�w~p�m�l�{Ywqs u¢}VxUm��zrV��u�jypVl��Uw~p"oqx³pkw~p"xz�zpkjyots�mns�oqpVln��.wqs�m/jy X}Votm��Uw~p"lb� � �7��O Ì �7�� 8O�N?���7��¯US���'���$��M`��� � ���ROa.?���7���'��O�� ù G&� <�D�K;�2<�<�CJ�D�C�C�CJ�

);:�DI, ü �tªJ�if�5jbm�xz�z��¨N��]³�zu��Um-lnjbm�l³{Ywqs³u¢}VxUm��zrV��u�jypVl��Uw~p"oqx�pkw~pVx��zpkjyots-mns�oqpVl��#wqs�m�jy �}"oc�m��Uw~pVly�*)O,7¥� ü�Oe�S94(>O�� ù G�@k� ù �$� K�<�:;�2K�:�\X�pD�C�CRDX�

);:�KI,Î] � ¥ �(ªXjb|qjyx´� ¥ m�ikjbwqs�jbm��z�boqx�lnm�}Vrk��wq{Ms�jy��jb�Vmnwqs�u�jy�iVoqpV�zl�u�l_��p­�"oq��mnjbs���oqx��iVjyu�wqm�oc¦J��lb��� Ì � +G)O � #R#��PO-+��� ��2O���K�D2� :�<�K;�2:�:�<X� ù @RG�G��

);:A+�,�ú¢�H´��-ªXw~xzx�oqpVrºú¢� ¬ jyl�lnjyx�lb��+B�� "� ��� � �����þ!A�$�'���6� È ��-/���&[$ûn�����þ�� ¬ �zxzjb�voqp"rªXw~pVly� � jb�b; wqs�¨N�NgPwqs�w~p�mnwk� ù @�@�\X�

);:�<I,�ú¢� ¬ �³ªJmns�w�wJ��¨#��ªXw~u�j�lnmnwJ��i"oqlnm��z�¢�Vs�wJ��jyl�lnjyl��/i"�z��i;ots��zlnj¢{Ys�w~u o�u�wXrkjyx³wq{m�ikj;u�wqm��Uw~p¡wq{�o �"oq��mnjbs�z}Vu��uUÀ*��©���©4Ow�w��ç��$!X�S����O-¶n�8���É�A�Î��D�\2� K�CR<;�2K ù <X�ù @RGI+k�

);:�:I, ¬ � ¬ jy�zl�lb�ÿ¯5w~pfm���p�}kw~}"l¢l�ikwX��¨3l�mns�}V��m�}ks�j��zp3j�¦JmnjypVrkjyr�m�ikjbsu�wXrV�XpVoqu@�z�blb�U��&!I��O��S��-AOw���%<�D2� <RG�:�CJ� ù @�@�<X�);:RG/,�ú¢�Hú¢� ¬ w�wJrk�hots�r-��´ �5g%�Xl�w~p-��þ �H´��(þ��qjbsl���w~}k|~i-�Y k�Hú¢��þ,}ks�s�oy�q� M �h�5}k�

rks�jypVjq� oqp"r ü � ¯��­�5jbs�|k� ªX�"otm��Uwt��mnjyu��.wqs�oqxR�"otmnmnjbspVl�|qjypkjbs�otmnjyr �X����������������'�P�Ï "!$#%������(2���(2��� ² � �$#%�&!I��O�)>Oz�%:�\2� D ù \ ù �2D ù \�@X� ù @�@�<X�

D�: