Numerical Simulation of Coexist between Host and Parasitoid for Improved Modification of Nicholson-Bailey Model

Rati Wongsathan Electrical Engineering Department, Faculty of Engineering North-Chiang Mai University, Chiang Mai 50230, Thailand

Tel: 0-5381-9999 Fax: 0-5381-9998 E-mail: chaiporn089@gmail.com, chaiporn@northcm.ac.th

Abstract—The improved modification of Nicholson-Bailey model for a two species host-parasitoid system with discrete generations assumes random distributions of both hosts and parasitoids, randomly searching parasitoids, and random encounters between the individuals of the two species that improved from [1] is presented. The linearization stability analysis showed that this model is stable on some region for the parameter values of reproductive rate of host and average number of viable eggs laid by a parasitoid on singing host. The numerical simulation results with various parameters showed that the steady state of host and parasitoid population density coexist depend on the fraction of the host population’s carrying capacity. Keywords-host/parasitoid model, Nicholson-Bailey model

I. INTRODUCTION Parasitoids are insect species which larvae develop as

parasites on other insect species. Parasitoid larvae usually kill its host (some times the host is paralyzed by ovipositing parasitoid female) whereas adult parasitoids are free-living insects species are either wasps or flies. Parasitoids and their hosts often have synchronized life-cycles, e.g., both have one generation per year (monovoltinous). Thus, host-parasite models usually use discrete time steps that correspond to generations (years).

A.J. Nicholson was one of the first biologists to suggest that host-parasitoid systems could be understood using a theoretical model, although only with the help of the physicist V.A. Bailey were his arguments given mathematical rigor [2]. The Nicholson-Bailey host-parasite model illustrates the properties of this model where search is at random and where searching efficiency is independent of both host and parasite density.

Nicholson followed Thomson (1924) by setting the simple model [3]:

)exp(1 ttt aPNN −=+ λ (1)

))exp(1(1 aPtNP tt −−=+ (2)

Where Nt,Nt+1,Pt, and Pt+1 are the density of host and parasite species in generation t and t+1 respectively. Parameter a is a constant, which represents the searching efficiency of parasite and λ is the increasing of host rate or reproductive rate of host. Such model have only one equilibrium, occurring when host and adult parasite

populations are equal to the steady densities ( N and P ). These densities depend on the values for a and λ [4]:

aN

)1(ln−

=λ

λλ (3)

aP λln= (4)

Linearization at this fixed point yields that the fixed point is unstable for all parameter values λ > 1 and a > 0.

Since the Nicholson-Bailey model is unstable for all parameters value, this conclusion spawned a growth industry of modeling attempting to show how the host-parasitoid interaction could be stabilized. In research paper [1] the modification of Nicholson-Bailey model are made by the following assumption:

(1) Hosts that have been parasitized will give rise to the next generation like parasitoids.

(2) Hosts have not bee parasitized will give rise to their own progeny.

(3) The fraction of hosts that are parasitized depends on the densities function of both species. We therefore define the following: ),( tt PNff = = fraction of hosts not parasitized, defined by

taP

tttt e

NEK

NEKPNf −

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 1),( (5)

where K is said to be the carrying capacity of the environment for the population or the limiting value of the population that can be supported in a particular environment, E is the fraction of the host population’s carrying capacity that can be accommodated in safe refuges then EK has units of population density and represent the maximum of population density of the refuges. c = average number of viable eggs which were laid by a parasitoid on single host. These three assumptions lead to:

taPtt eEKNEKN −

+ −+= )(1 λλ (6)

)1)((1taP

tt eEKNcP −+ −−= (7)

2009 International Conference on Signal Processing Systems

978-0-7695-3654-5/09 $25.00 © 2009 IEEE

DOI 10.1109/ICSPS.2009.124

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After analysis the stability, the equilibrium point of host and parasitoid population density are found that [1]

EKca

N +≈ 1 (8)

aEKcaP

λλ )1)(1( +−≈ (9)

To stabilize the model, then the implicit analysis was made by using simulation plot on λc-parameter plane by fixing the value of parameter E, K and a. The example of plotting is shown by Fig. 1.

Figure 1. Stable region for fixed value of E=0.35,

K=50, and a=0.02

To make host parasitoid model more realistic then followed by this section, we described the improve model in section II. Abbreviation of derivation on stability analysis is done in section III. Results and discussion are discussed in section IV. Finally, conclusion is summarized in section V.

II. IMPROVED MODIFICATION OF HOST AND PARASITOID MODEL

If f(Nt,Pt) is the fraction of the host population that is not parasited; then this must be equal to the fraction that are safe in refuges plus the fraction there were not in refuges but still managed to escape attack. The fraction of the population that can retreat to a refuge in generation t is EK/Nt and the fraction that is not safe is then 1-EK/Nt. Those that are not safe have probably exp(-aPt) of escaping attack. Taken together, the fraction of host not parasitized is defined in (5) as same as [1]. But the difference between our model and [1] is the model of host population that defines:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+= −⎟

⎠⎞

⎜⎝⎛ −

+t

taP

ttt

KN

t eNEK

NEKNeN 1

1

1

λ (10)

Consider the exponential growth, it depend on parameter λ and Nt. In the case of λ kept constant while vary on Nt, we see that if Nt = 0 (mean host can increase the population to carrying capacity value), exponential term with λ has higher

value, if Nt reach to K (mean host’s population are saturate) then exponential term with λ has lower value. This relation is shown by Fig. 2.

Figure 2. Relation between Nt and exponential grow.

The equation for the parasitoid population should state that the number of parasitoid in the next generation is equal to the number of hosts in the previous generation that were not parasitized multiplied by the number of viable eggs per host. This translates to:

]1)[(1taP

tt eEKNcP −+ −−= (11)

III. STABILITY ANALYSIS To check the stability, first we try to find the steady state

of host and parasitoid population. Consider (11) we place Pt+1 and Pt by P and Nt by N and make a little algebra then we get

0=P (12) and

EK

cPaaN +

−≈

)2

(

12

(13)

Replace Nt+1 and Nt in (10) to N and Pt to P , then place P in (12) into (10) to find value that correspond to N . It’s solve easily then we found KN = (14) For the same manner, we place N in (13) into (10), and use Taylor series to expand exponential term, then we get

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−−−=

NEK

NEK

KN

aP

1

)1exp(ln1 (15)

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After calculation is made, we receive the 2 stables of ( )PN , for this model:

( )⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−

NEK

NEK

KN

aEKPcacaK

1

)1exp(ln1,

2,0,

12

For stable 1, we can use this point to calculate the matrix of J:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−−=)(0

)(1

EKKca

KEKaKN

Jλ

The condition for stability is:

)(1))()(1(12 EKKcaEKKca −+−<−−+< λλ

That is very complicated and difficult to analyze explicitly then we analyze by using simulation plot on λc- parameter plane by fixing the value of parameter E, K and a then vary on parameter λ and c to investigate the stability of host and parasitoid population.

Figure 3. Stable region of λ and c for fixed E =0.1

From Fig.3, we fixed the value of E=0.1, a=0.02 and K=50 then the region of stability showed that c is less than 1.1 and λ is less 2. If we vary E to 0.1, 0.2 and 0.3 then the stable area was shown in Fig. 4.

We see that for higher value of E that’s mean the host capable to refuge more then the parasitoid will increase their population by add more egg. In opposite, if E is lower that’s mean host has less capable to refuge then the parasitoid will decrease to laid egg for preserving host in the next generation.

For stable 2, it’s too difficult and complicate to obtain the matrix J for explicitly by analytical solve then we use the trial and error. It was found that the stable region has the same interval of λ like as stable 1 but interval of c is the compliment in stable 1. We can show the numerical solution plots of variety cases in the next section.

Figure 4. Stable region of λ and c for variety of E

IV. NUMERICAL SIMULATION RESULTS AND DISCUSSION In this section, we showed the numerical solution plot

between host and parasite population in each generation. We started with the stable 1 which fixed a=0.02, K=50 and use stable region for λ =1 and c=0.5 with variety of E=0.1 and 0.2. The results were shown in Fig. 5 and 6, respectively.

Figure 5. Host and parasite approach to the steady state with

a = 0.02, K = 50, λ = 1, c = 0.5 and E = 0.1 for stable 1.

Figure 6. Host and parasite approach to the steady state with

a = 0.02, K = 50, λ = 1, c = 0.5 and E = 0.2 for stable 1.

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For λ = 2 with the same parameter of a, K and c with variety of E = 0.1 and 0.2, the results were shown in Fig. 7 and 8, respectively.

Figure 7. Host and parasite approach to the steady state with a = 0.02, K = 50, λ = 2, c = 0.5 and E = 0.1 for stable 1.

Figure 8. Host and parasite approach to the steady state with a = 0.02, K = 50, λ = 2, c = 0.5 and E = 0.2 for stable 1.

Numerical simulations of the model show that when there are no parasitoids (P = 0), the host population will monotonically approach the environmental carrying capacity as long as λ is less than a threshold value. As λ increases ( λ = 2), the host population will eventually exhibit chaotic behavior as a result of period-doubling bifurcations. For stable 2, we tested with the same manner and same parameters except for parameter c that we used about 3 for compliment of the stable 1, the results were shown in Fig.9, 10, 11 and 12, respectively. We see that for the suitable parameters λ and c which we selected form the stability regions in Fig. 4, the population density of host and parasitoid are fluctuated in the beginning generation and

then kept stable in the next few generations and converged to the stable points ( )PN , .

Figure 9. Host and parasite approach to the steady state with a = 0.02, K = 50, λ = 1, c = 3 and E = 0.1 for stable 2.

Figure 10. Host and parasite approach to the steady state with a = 0.02, K = 50, λ = 1, c = 3 and E = 0.2 for stable 2.

Figure 11. Host and parasite approach to the steady state with a = 0.02, K = 50, λ = 2, c = 3 and E = 0.1 for stable 2.

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Figure 12. Host and parasite approach to the steady state with

a = 0.02, K = 50, λ = 2, c = 3 and E = 0.2 for stable 2. When there are parasitoids present, the existence of refuges has a stabilizing effect on both the populations. For E small, both populations coexist and approach their respective steady states via damped oscillations. As E increases in Fig. 13, the parasitoid population dies out and the host approach to the environmental carrying capacity.

Figure 13. Host and parasite approach to the steady state with

a = 0.02, K = 50, λ = 2, c = 3 and E = 0.4 for stable 2.

V. CONCLUSION The improved modification of Nicholson-Bailey for host-

parasitoid model is more realistic than model [1].The steady-state of host and parasitoid population has two points which stable in some region of cλ -plane. The numerical solution for various parameters showed that when there are no parasitoids, the host population will monotonically approach the environmental carrying capacity as long as λ is less than a threshold value. As λ increases, the host population will eventually exhibit chaotic behavior as a result of period-doubling bifurcations. When there are parasitoids present, the existence of refuges has a stabilizing effect on both the populations. For E small, both populations coexist and approach their respective steady states via damped oscillations. As E increases, the parasitoid population dies out and the host approach to the environmental carrying capacity.

REFERENCES

[1] R. Wongsathan, “Stability Analysis on Modification of Nicholson-Bailey Host-Parasitoid System,” Scinece Research Conf 2nd, Feb, 2009.

[2] S. Kingsland, “Modeling Nature: Episodes in the History of Population Ecology,” University of Chicago Press, Chicago, 1985.

[3] V.A. Bailey, A.J. Nicholson, and E.J. Williams, “Interaction between hosts and parasites when some host individuals are more difficult to find than others,” J. theor. Biol. 3, pp. 1-18, 1962.

[4] M.P Hassell, and R.M. May, "Stability in Insect Host-Parasite Models," Ani. Eco. J., Vol. 43, No. 3, pp. 693-726, 1973.

[5] Leah Edelstein-Keshet, Mathematical Models in Biology, Duke University, New York, USA.

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