on predator-prey modelsjxshix.people.wm.edu/math-410-2010/2010-csums.pdf · on predator-prey models...
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Predator-prey
Junping Shi
On predator-prey models
Junping Shi $�²
Department of Mathematics
College of William and Mary
Math 410/CSUMS TalkFebruary 3, 2010
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Collaborators
Sze-Bi Hsu (Tsinghua University, Hsinchu, Taiwan)
Junjie Wei (Harbin Institute of Technology, Harbin/Weihai,China)
Rui Peng (University of New England, Armidale, Australia)
Fengqi Yi (Harbin Engineering University, Harbin, China)Ph.D. 2008
Jinfeng Wang (Harbin Institute of Technology/HarbinNormal University, Harbin, China)Ph.D. expected 2010
Yuanyuan Liu (William & Mary, Williamsburg, USA)B.S. expected (math and econ) 2010
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Fur trade of Hudson Bay Company (1670-1950)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Hudson Bay Company lynx-hare data
Charles Elton (1924), “Periodic fluctuations In the numbersof animals: their causes and effects”, British Journal of
Experimental Biology , was first (of MANY) publications toanalyze this data setRebecca Tyson, et.al. (2009), “Modelling the Canada lynxand snowshoe hare population cycle: the role of specialistpredators”, Theoretical Ecology , one of the latest.
Predator-prey
Junping Shi
Lotka-Volterra Predator-Prey Model
Alfred Lotka (1880-1949) Vito Volterra (1860-1940)
du
dt= au − buv ,
dv
dt= −cv + duv .
see Math 302 Chapter 2http://en.wikipedia.org/wiki/Lotka-Volterra_equation
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Predator-prey system with functional response
du
dt= u(a − bu) − cφ(u)v ,
dv
dt= −dv + f φ(u)v .
φ(u): predator functional responseφ(u) = u (Lotka-Volterra)
φ(u) =u
1 + mu(Holling type II, m: prey handling time)
[Holling, 1959](Michaelis-Menton biochemical kinetics)
Biological work:[Rosenzweig-MacArthur, American Naturalist 1963][Rosenzweig, Science, 1971] (Paradox of enrichment)[May, Science, 1972] (Existence/uniqueness of limit cycle)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Rosenzweig-MacArthur model
du
dt= u
(1 −
u
K
)−
muv
1 + u,
dv
dt= −dv +
muv
1 + u
Nullcline: u = 0, v =(K − u)(1 + u)
m; v = 0, d =
mu
1 + u.
Solving d =mu
1 + u, one have u = λ ≡
d
m − d.
Equilibria: (0, 0), (1, 0), (λ, vλ) where vλ =(K − λ)(1 + λ)
mWe take λ as a bifurcation parameter
Case 1: λ ≥ K : (K , 0) is globally asymptotically stableCase 2: (K − 1)/2 < λ < K : (K , 0) is a saddle, and (λ, vλ)is a locally stable equilibriumCase 3: 0 < λ < (K − 1)/2: (K , 0) is a saddle, and (λ, vλ) isa locally unstable equilibrium (λ = (K − 1)/2 is a Hopfbifurcation point)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Phase Portraits
Left: (K − 1)/2 < λ < K : (K , 0) is a saddle, and (λ, vλ) is alocally stable equilibriumRight: 0 < λ < (K − 1)/2: (K , 0) is a saddle, and (λ, vλ) isa locally unstable equilibrium; there exists a limit cycle
A supercritical Hopf bifurcation occurs.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Global stability
[Hsu, Hubble, Waltman, SIAM J. Appl. Math., 1978][Hsu, Math. Biosci., 1978] (λ, vλ) is globally asymptoticallystable if K ≤ 1, or K > 1 and (K − 1)/2 < λ < K .
[Cheng, SIAM J. Math. Anal., 1981] If 0 < λ < (K − 1)/2,then (λ, vλ) is unstable, and there is a unique periodic orbitwhich is globally asymptotically stable.
More on uniqueness of limit cycle:[Zhang, 1986], [Kuang-Freedman, 1988][Hsu-Hwang, 1995,1998], [Xiao-Zhang, 2003, 2008]
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Summary of ODE
du
dt= u
(1 −
u
K
)−
muv
1 + u,
dv
dt= −dv +
muv
1 + u
Nullcline: u = 0, v =(K − u)(1 + u)
m; v = 0, d =
mu
1 + u.
Solving d =mu
1 + u, one have u = λ ≡
d
m − d.
Equilibria: (0, 0), (K , 0), (λ, vλ) where vλ =(K − λ)(1 + λ)
mWe take λ as a bifurcation parameter
Case 1: λ ≥ K : (K , 0) is globally asymptotically stableCase 2: (K − 1)/2 < λ < K : (λ, vλ) is globallyasymptotically stableCase 3: 0 < λ < (K − 1)/2: unique limit cycle is globallyasymptotically stable (λ = (K − 1)/2: Hopf bifurcationpoint)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Phase portrait (1)du
dt= u
(1 −
u
K
)−
muv
1 + u,
dv
dt= −dv +
muv
1 + u
Case 1: λ ≥ K : (K , 0) is globally asymptotically stable
u(0,0) (k,0)
v
(0,0) (k,0)
v=f(u)
• •
d=g(u)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Phase portrait (2)du
dt= u
(1 −
u
K
)−
muv
1 + u,
dv
dt= −dv +
muv
1 + u
Case 2: (K − 1)/2 < λ < K : (K , 0) is a saddle, and (λ, vλ)is a locally stable equilibrium
u(0,0) (k,0)
v
(0,0) (k,0)
v=f(u)
• •
d=g(u)
•(λ,f(λ))
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Phase portrait (3)du
dt= u
(1 −
u
K
)−
muv
1 + u,
dv
dt= −dv +
muv
1 + u
Case 3: 0 < λ < (K − 1)/2: (K , 0) is a saddle, and (λ, vλ) isa locally unstable equilibrium
u(0,0) (k,0)
v
(0,0) (k,0)
v=f(u)
• •
d=g(u)
• (λ,f(λ))
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
A new result of this ODE
[Hsu-Shi, Disc. Cont. Dyna. Syst.-B, 2009]Relaxation oscillator profile of limit cycle in predator-preysystem. (Motivated by numerical observation)du
dt= u (1 − u) −
muv
a + u,
dv
dt= −dv +
muv
a + u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
u
v
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Graph of limit cycle
Parameters: a = 0.5, m = 1, d = 0.1, λ = 1/18 ≈ 0.056,period T ≈ 37.
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
u an
d v
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Small d
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
u
v
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Graph of limit cycle
Parameters: a = 0.5, m = 1, d = 0.01, λ = 1/198 ≈ 0.005,period T ≈ 336.
0 100 200 300 400 500 600 700 800 900
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t
u an
d v
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Illustration of limit cyclev
O1
O2 O3
O4
v4(u)
v6(u)
v5(u)
a/m O5
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Relaxation oscillation
Theorem[Hsu-Shi, 2009] If 0 < a < 1 and m > 0 are fixed,and as d → 0 (thus λ → 0), thenC1λ
−1 ≤ T (O1O2) ≤ C2λ−1, T (O2O3) = O(| lnλ|),
T (O4O1) = O(| lnλ|), and T (O3O4) = O(1). In particular,the period T → ∞ as d → 0. The shape of the graph of thelimit cycle is a relaxation oscillator.
Other known relaxation oscillators:Van der Pol oscillator in electrical circuitsFitzHugh-Nagumo oscillator in action potentials of neuronsMany other physiology models: heart beat, calcium signaling
[Liu-Xiao-Yi, JDE, 2003] oscillations in singularly perturbed3-D predator-prey system[Zhang-Wang-Wang-Shi, preprint] extend it to more generalpredator-prey system
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Relaxation oscillation
(left) Van der Pol oscillator;(right) FitzHugh-Nagumo oscillator.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
More general predator-prey system
du
dt= g(u) (f (u) − v) ,
dv
dt= v (g(u) − d) .
(a1) f ∈ C 3(R+), f (0) > 0, there exists K > 0, such thatfor any u > 0, u 6= K , f (u)(u − K ) < 0 and f (K ) = 0; thereexists λ̄ ∈ (0,K ) such that f ′(u) > 0 on [0, λ̄), f ′(u) < 0 on(λ̄,K ];(a2) g ∈ C 2(R+), g(0) = 0; g(u) > 0 for u > 0 andg ′(u) > 0 for u ≥ 0; there exists a unique λ ∈ (0,K ) suchthat g(λ) = d .
g(u): functional response, f (u): prey isocline
Rosenzweig-MacArthur model: g(u) =mu
1 + uand
f (u) =(K − u)(1 + u)
m
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Phase portraits
du
dt= g(u) (f (u) − v) ,
dv
dt= v (g(u) − d) .
u(0,0) (k,0)
v
(0,0) (k,0)
v=f(u)
• •
d=g(u)
•(λ,f(λ))
u(0,0) (k,0)
v
(0,0) (k,0)
v=f(u)
• •
d=g(u)
• (λ,f(λ))
Left: (λ, f (λ)) stable;Right: (λ, f (λ)) unstable, exist limit cycle
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
History
du
dt= g(u) (f (u) − v) ,
dv
dt= v (g(u) − d) .
[Hsu, 1979] If λ > λ̄ and f (u) is concave in [0,K ], then(λ, f (λ)) is globally stable. (λ̄ is the top of the hump)
[Kuang-Freedman, 1988] If 0 < λ < λ̄, andd
du
(f ′(u)g(u)
g(u) − d
)≤ 0 for all x ∈ [0,K ], then the limit cycle
is unique and globally stable.
[Hofbauer-So, 1989] counterexample: f (u) is concave, butHopf bifurcation is subcritical, so there are two periodicorbits for λ ∈ (λ̄, λ̄ + ǫ), one of them is locally stable.
More work:[Ruan-Xiao, 2001] [Xiao-Zhang, 2003] [Liu, 2005]
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Our resultdu
dt= g(u) (f (u) − v) ,
dv
dt= v (g(u) − d) .
Theorem[Wang-Shi-Wei, submitted] If λ > λ̄, and(a8) (uf ′(u))′′ ≤ 0, (u/g(u))′′ ≥ 0 for u ∈ [0,K ], and(uf ′(u))′ ≤ 0 for u ∈ (λ̄,K ); or(a9) f ′′′(u) ≤ 0, (1/g(u))′′ ≥ 0 for u ∈ [0,K ], andf ′′(u) ≤ 0 for u ∈ (λ̄,K ),then (λ, f (λ)) is globally stable.
Sharpness: If (uf ′(u))′′|u=λ̄> 0 and g(u) = u/(a + u) or
g(u) = u, then Hopf bifurcation is subcritical and globalstability does not hold.
Realistic example:du
dt= ru
(1 −
u
K
)(1 −
A + C
u + C
)−Buv ,
dv
dt= −dv + Buv .
Prey growth: weak Allee effect when A < 0 and C > −A.C large: global stability;C small: subcritical Hopf bifurcation.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Global stability vs. Multiple periodic orbits
du
dt= ru
(1 −
u
K
)(1 −
A + C
u + C
)−Buv ,
dv
dt= −dv + Buv .
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
u
v
•
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
u
v•
Both: r = B = 1, A = −0.028, K = 1;(left) d = 0.10199, C = 0.05; (subcritical Hopf)(right) d = 0.6, C = 2 (supercritical Hopf).
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Habitat for predator and prey
Habitat fragmentation
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Habitat Restoration: connecting patches
Crossing bridges built in Banff, Canada; (for elks)under-cross built in Florida, USA (for deers)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Habitat network
Habitat with a network structure
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Habitats as graph theory
Network connectivity
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Effect of network structure on dynamics
Matthew Holland, Alan Hastings, (2008), “Strong effect ofdispersal network structure on ecological dynamics”, Nature.
David Vasseur, Jeremy Fox, (2009), “Phase-locking andenvironmental fluctuations generate synchrony in apredator‘prey community”, Nature.
dNi
dt= Ni
(1 −
Ni
K
)−
mNiPi
1 + Ni+ dn
n∑
j=1
dij(Nj − Ni ),
dPi
dt=
mNiPi
1 + Ni− ePi + dp
n∑
j=1
dij(Pj − Pi ).
Homogeneous patches, dispersal matrix (dij) (dij = dji)single patch: Rosenzweig–MacArthur model
[Holland-Hastings]: n = 10, random network; clustersolutions, long transient dynamics.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Stability in network model
Local dynamics:du
dt= u
(1 −
u
K
)−
muv
1 + u,
dv
dt= −ev +
muv
1 + u
Case 1: λ ≥ K : (K , 0) is globally stableCase 2: (K − 1)/2 < λ < K : (λ, vλ) is globally stableCase 3: 0 < λ < (K − 1)/2: unique limit cycle is globallystable (λ = (K − 1)/2: Hopf bifurcation point)
Theorem[Wang-Shi-Liu, in preparation]If λ ≥ K , then (K , 0)n is globally stable;and if K − 1 < λ < K , then (λ, vλ)n is globally stable.
[Li-Shuai (2010), JDE] Lyapunov funct. ODE on network
Note: same result holds for reaction-diffusion model.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Two-patch model
u′ = F (u, v) + a(w − u),
v ′ = G (u, v) + c(x − v),
w ′ = F (w , x) − a(w − u),
x ′ = G (w , x) − c(x − v),
where
F (u, v) = u(1 −
u
K
)−
muv
1 + u, G (u, v) =
muv
1 + u− ev .
where a, c > 0 are the diffusion rates.Can we completely classify the dynamics?
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Some Partial Results for Two-patch model
[Liu-Shi, in preparation]Let U(t) = u(t) + w(t), V (t) = v(t) + x(t) (sum)and W (t) = u(t) − w(t), X (t) = v(t) − x(t) (difference)
1. When a, c are large, then W (t), X (t) → 0 as t → ∞ so thesystem is synchronized.2. For 0 < a < a0, there exists c0 = c0(a) such that when0 ≤ c < c0, there are two additional Hopf bifurcation pointsλ−
H , λ+H , and there exists a non-symmetrical periodic orbit for
λ ∈ (λ−
H , λ+H). (numerical result shows the non-symmetrical
periodic orbit is unique and unstable)
3. For 0 ≤ a < a1, there exists c1 = c1(a) such that when c > c1,
there are two additional bifurcation points λ−
S , λ+S such that there
exists two non-symmetrical equilibrium points for λ ∈ (λ−
S , λ+S ).
((a) equilibrium points can be algebraically solved with a
complicated formula; (b) for any a ≥ 0 and c ≥ 0, there are at
most 9 equilibrium points.)
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Numerical bifurcation diagrams
Software: Matlab and MatCont (Govaerts, Kuznetsov)similar to Auto, but works under MatlabBifurcation/continuation of equilibrium, limit cycles,homoclinic orbits of ODEs
−0.1 0 0.1 0.2 0.3 0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
e
Max
(u)
H
H
H
H
H
H BP
BP
LPCLPCLPCLPCLPCLPCLPCLPC
k = 4, m = 0.5, a = 0.06, c = 0.01, bifurcation parameter 0 < e < 0.4;
symmetric Hopf: e = 0.3, non-symmetric Hopf: e = 0.2387 and e = 0.1173
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Numerical solution
Software: Matlab
0 50 100 1500
1
2
3
4
5
6
time
coup
led
popu
latio
n
prey−1predator−1prey−2predator−2
k = 4, m = 0.5, a = 0.06, c = 0.01, bifurcation parameter 0 < e < 0.4;
symmetric Hopf: e = 0.3, non-symmetric Hopf: e = 0.2387 and e = 0.1173
Here e = 0.2
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Numerical solution
Software: Matlab
0 100 200 300 400 500 600 700 800 900 1000−2
0
2
4
time
patc
h−1
popu
latio
n
0 100 200 300 400 500 600 700 800 900 1000−2
0
2
4
time
patc
h−2
popu
latio
n
0 100 200 300 400 500 600 700 800 900 10000
1
2
3
time
coup
led
popu
latio
n
0 100 200 300 400 500 600 700 800 900 10000
1
2
3
time
coup
led
popu
latio
n
prey−1predator−1
prey−2predator−2
prey−1predator−1prey−2predator−2
total prey populationtotal predator population
k = 5, m = 9.96, a = 0, c = 0.6, bifurcation parameter 0 < e < 10;
symmetric Hopf: e = 6.64, no non-symmetric Hopf
Here e = 1 and solution appears to be chaotic
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Attractor? Chaos?
An attractor exists as all positive solutions are bounded.
When there are non-symmetrical equilibrium points (EQ) orperiodic orbits (PO), there is only one stable state(symmetrical PO) in the attractor, but more than 2 unstablestates (symmetric EQ, non-symmetric EQ and/or PO).
Hence there exist connecting orbits between the two unstablestates, which become a heteroclinic loop. This may implychaotic behavior on a lower dimensional invariant manifold.
Conjecture: Chaos occurs on a zero measure set, but thegeneric dynamics is eventual synchronization (despitepossible long transient dynamics) and converges tosymmetric PO.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Non-symmetric patches (spatial heterogeneity)
u′ = F1(u, v) + a(w − u),
v ′ = G1(u, v) + c(x − v),
w ′ = F2(w , x) − a(w − u),
x ′ = G2(w , x) − c(x − v),
where for i = 1, 2,
Fi (u, v) = u
(1 −
u
Ki
)−
miuv
1 + u, Gi(u, v) =
miuv
1 + u− eiv .
where a, c > 0 are the diffusion rates.
1. synchronization of coupled oscillators2. synchronization of equilibrium and oscillatorGoldwyn, E.E. and Hastings, A. (2008) When can dispersal synchronize populations? Theoretical
Population Biology 73:395-402
Goldwyn, E.E. and Hastings, A. (2009) Small heterogeneity has large effects on synchronization of
ecological oscillators. Bulletin of Mathematical Biology 71:130-144.
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Non-symmetric patches: numerical results
Y.Y. Liu (undergraduate research)x0 = [0.95; 0.05; 0.2; 0.5]; k = 3; m = 2; e1 = 0.2; e2 = 0.9Left: a = 0.1; c = 0.2; Right: a = 1; c = 2.
0 10 20 30 40 50 60 70 80 90 1000
2
4
time
patc
h−1
popu
latio
n
prey−1predator−1
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
time
patc
h−2
popu
latio
n
prey−2predator−2
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
time
coup
led
popu
latio
n
prey−1predator−1prey−2predator−2
0 10 20 30 40 50 60 70 80 90 1000
2
4
time
coup
led
popu
latio
n
total prey populationtotal predator population
0 10 20 30 40 50 60 70 80 90 1000
2
4
time
patc
h−1
popu
latio
n
prey−1predator−1
0 10 20 30 40 50 60 70 80 90 1000
2
4
time
patc
h−2
popu
latio
n
prey−2predator−2
0 10 20 30 40 50 60 70 80 90 1000
2
4
time
coup
led
popu
latio
n
prey−1predator−1prey−2predator−2
0 10 20 30 40 50 60 70 80 90 1000
2
4
time
coup
led
popu
latio
n
total prey populationtotal predator population
Larger diffusion rates help to achieve synchronization;smaller diffusion rates may not result in synchronization
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Non-symmetric patches: numerical results
Y.Y. Liu (undergraduate research)x0 = [0.95; 0.05; 0.2; 0.5]; k = 3; m = 2; e1 = 1.8; e2 = 0.45Left: a = 0.1; c = 0.2; Right: a = 1; c = 2.
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
patc
h−1
popu
latio
n
prey−1predator−1
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
patc
h−2
popu
latio
n
prey−2predator−2
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
coup
led
popu
latio
n
prey−1predator−1prey−2predator−2
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
coup
led
popu
latio
n
total prey populationtotal predator population
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
patc
h−1
popu
latio
n
prey−1predator−1
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
patc
h−2
popu
latio
n
prey−2predator−2
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
coup
led
popu
latio
n
prey−1predator−1prey−2predator−2
0 50 100 150 200 250 300 350 400 450 5000
2
4
time
coup
led
popu
latio
n
total prey populationtotal predator population
Connecting a prey-only (predator extinction) system to anoscillatory system could make a stable coexistence
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
Non-symmetric patches: numerical results
x0 = [0.95; 0.05; 0.2; 0.5]; k = 3; m = 2; e1 = 0.2; e2 = 0.9Left: a = 0.1; c = 0.2
The limit cycle is non-symmetrical, and not synchronized.
0 50 100 150 200 250 300−2
0
2
4
time
patc
h−1
popu
latio
n
0 50 100 150 200 250 3000
1
2
3
time
patc
h−2
popu
latio
n
0 50 100 150 200 250 3000
1
2
3
time
coup
led
popu
latio
n
0 50 100 150 200 250 3000
1
2
3
4
time
coup
led
popu
latio
n
prey−1predator−1
prey−2predator−2
prey−1predator−1prey−2predator−2
total prey populationtotal predator population
Predator-prey
Junping Shi
Future Work (for CSUMS)
1. Further work on 2-patch model (complete bifurcationdiagram)2. Impact or network structure (small number of patches):3-patch (linear or triangle)3. If there is a cycle in the network, is there a periodicsolution with population running around the cycle?4. Computer work: writing better Matlab programs forbifurcation diagram calculation
spatial
predator-prey
systems
Junping Shi
Background
ODE Model
Limit Cycle Profile
Subcritical Hopf
Connected Patches
Reaction-Diffusion
Conclusion
References
◮ Sze-Bi Hsu; Junping Shi Relaxation oscillator profile of
limit cycle in predator-prey system. Discrete and
Continuous Dynamical Systems B, 11, (2009) no. 4,893–911.
◮ Fengqi Yi, Junjie Wei and Junping Shi, Bifurcation and
spatiotemporal patterns in a homogeneous diffusive
predator-prey system. Journal of Differential Equations,246, (2009), no. 5, 1944–1977.
◮ Rui Peng, Junping Shi, Non-existence of Non-constant
Positive Steady States of Two Holling Type-II
Predator-prey Systems: Strong Interaction Case.Journal of Differential Equations, 247, (2009), no. 3,866–886.
Predator-prey
Junping Shi