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Mathematical Problems in Engineering
Volume 2008, Article ID 945420, 8 pages
http://dx.doi.org/10.1155/2008/945420
Research Article

Solving Ratio-Dependent Predator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method

1Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman 76169, Iran
2Department of Civil and Mechanical Engineering, Mazandaran University of Technology, P.O. Box 484, Babol 47144, Iran

Received 1 February 2008; Revised 29 February 2008; Accepted 13 March 2008

Academic Editor: Cristian Toma

Copyright © 2008 Abdoul R. Ghotbi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Due to wide range of interest in use of bioeconomic models to gain insight into the scientific management of renewable resources like fisheries and forestry, homotopy perturbation method is employed to approximate the solution of the ratio-dependent predator-prey system with constant effort prey harvesting. The results are compared with the results obtained by Adomian decomposition method. The results show that, in new model, there are less computations needed in comparison to Adomian decomposition method.

1. Introduction

Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly, and even if an exact solution is obtainable, the required calculations may be practically too complicated, or it might be difficult to interpret the outcome. Very recently, some promising approximate analytical solutions are proposed such as Exp-function method, Adomian decomposition method (ADM), variational iteration method (VIM), and homotopy perturbation method (HPM).

HPM is the most effective and convenient method for both linear and nonlinear equations. This method does not depend on a small parameter. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter 𝑝[0,1], which is considered as a “small parameter.” HPM has been shown to effectively, easily, and accurately solve a large class of linear and nonlinear problems with components converging to accurate solutions. HPM was first proposed by He [17] and was successfully applied to various engineering problems.

The motivation of this paper is to extend the homotopy perturbation method (HPM) [817] to solve the ratio-dependent predator-prey system. The results of HPM are compared with those obtained by the ADM [18]. Different from ADM, where specific algorithms are usually used to determine the Adomian polynomials, HPM handles linear and nonlinear problems in simple manner by deforming a difficult problem into a simple one. The HPM is useful to obtain exact and approximate solutions of linear and nonlinear differential equations.

In this paper, we assume that the predator in model is not of commercial importance. The prey is subjected to constant effort harvesting with 𝑟, a parameter that measures the effort being spent by a harvesting agency. The harvesting activity does not affect the predator population directly. It is obvious that the harvesting activity does reduce the predator population indirectly by reducing the availability of the prey to the predator. Adopting a simple logistic growth for prey population with 𝑒>0,𝑏>0, and 𝑐>0 standing for the predator death rate, capturing rate, and conversion rate, respectively, we formulate the problem as𝑑𝑥𝑑𝑡=𝑥(1𝑥)𝑏𝑥𝑦𝑦+𝑥𝑟𝑥,𝑑𝑦=𝑑𝑡𝑐𝑥𝑦𝑦+𝑥𝑒𝑦,(1.1) where 𝑥(𝑡) and 𝑦(𝑡) represent the fractions of population densities for prey and predator at time 𝑡, respectively. Equations (1.1) are to be solved according to biologically meaningful initial conditions 𝑥(0)0 and 𝑦(0)0 [18].

2. Applications

In this section, we will apply the HPM to nonlinear differential system of ratio-dependant predator-prey,𝐻=𝐿𝜈𝑢𝜈,𝑝1𝑝𝐿0𝐴𝜈𝑟+𝑝𝑓=0,𝑝0,1,𝑟𝜀Ω,(2.1)where 𝐴(𝜈) is a general differential operator which can be divided into a linear part 𝐿(𝜈) and a nonlinear part 𝑁(𝜈) and 𝑓(𝑟) is a known analytical function. 𝑝[0,1] is an embedding parameter, while 𝑢0 is an initial approximation of the equation which should be solved, and satisfies the boundary conditions.

According to the HPM (relation (2.1)), we can construct a homotopy of system as follows: 𝜈1𝑝2̇𝜈1+𝜈1̇𝜈1̇𝑥0y0̇𝑥0𝑥0𝜈+𝑝2̇𝜈1+𝜈1̇𝜈1𝜈1𝑏𝑟1𝜈2+𝜈2𝜈21𝜈1𝑟21+𝜈31×𝜈=0,1𝑝2̇𝜈2+𝜈1̇𝑣2̇y0y0x0̇y0𝜈+𝑝2̇𝜈2+𝜈1̇𝜈2+𝜈𝑒𝑐1𝜈2+e𝜈22=0,(2.2) where dot denotes differentiation with respect to 𝑡, and the initial approximations are as follows:𝑣1,0(𝑡)=𝑥0𝑣(𝑡)=𝑥(0),2,0(𝑡)=𝑦0(𝑡)=𝑦(0).(2.3) Assume that the solution of (2.2) can be written as a power series in 𝑝 as follows:𝜈1=𝜈1,0+𝑝𝜈1,1+𝑝2𝜈1,2+𝑝3𝜈1,3𝜈+,2=𝜈2,0+𝑝𝜈2,1+𝑝2𝜈2,2+𝑝3𝜈2,3+,(2.4) where 𝜈𝑖,𝑗(𝑖,𝑗=1,2,3,) are functions yet to be determined. Substituting (2.3) and (2.4) into (2.2), and arranging the coefficients of p powers, we have𝑣2,0̇𝑣1,0+𝑣1,0̇𝑣1,0+𝑣31,0𝑣21,0+𝑣1,0̇𝑣1,1+𝑣2,0̇𝑣1,1+𝑟𝑣1,0𝑣2,0+𝑏𝑣1,0𝑣2,0𝑣1,0𝑣2,0+𝑣2,0𝑣21,0+𝑟𝑣21,0𝑝+𝑣1,1̇𝑣1,1+𝑣1,0̇𝑣1,2+𝑣2,0̇𝑣1,2+𝑣2,1̇𝑣1,1+2𝑟𝑣1,0𝑣1,1+𝑏𝑣1,0𝑣2,1+2𝑣2,0𝑣1,0𝑣1,1+𝑟𝑣1,1𝑣2,0+𝑟𝑣1,0𝑣2,1+𝑏𝑣1,1𝑣2,0𝑣1,0𝑣2,1𝑣1,1𝑣2,0+𝑣2,1𝑣21,02𝑣1,0𝑣1,1+3𝑣21,0𝑣1,1𝑝2+𝑣1,1̇𝑣1,2+𝑣1,2̇𝑣1,1+𝑣1,0̇𝑣1,3+𝑣2,1̇𝑣1,2+𝑣2,0̇𝑣1,3+𝑣2,2̇𝑣1,1+𝑣2,0𝑣21,1𝑣1,0𝑣2,2𝑣1,2𝑣2,0𝑣1,1𝑣2,1+𝑣2,2𝑣21,0+𝑟𝑣21,1+3𝑣1,0𝑣21,1𝑣21,1+𝑏𝑣1,1𝑣2,1+𝑏𝑣1,0𝑣2,2+𝑏𝑣1,2𝑣2,0+𝑟𝑣1,0𝑣2,2+𝑟𝑣1,1𝑣2,1+𝑟𝑣1,2𝑣2,0+2𝑣2,0𝑣1,0𝑣1,2+2𝑟𝑣1,0𝑣1,2+2𝑣2,1𝑣1,0𝑣1,1+3𝑣21,0𝑣1,22𝑣1,0𝑣1,2𝑝3𝑣+=0,2,0̇𝑣2,0+𝑣1,0̇𝑣2,0+𝑒𝑣1,0𝑣2,0𝑐𝑣1,0𝑣2,0+𝑣2,0̇𝑣2,1+𝑣1,0̇𝑣2,1+𝑒𝑣22,0𝑝+𝑣2,1̇𝑣2,1+𝑒𝑣1,0𝑣2,1𝑐𝑣1,0𝑣2,1+𝑒𝑣1,1𝑣2,0𝑐𝑣1,1𝑣2,0+2𝑒𝑣2,0𝑣2,1+𝑣2,0̇𝑣2,2+𝑣1,1̇𝑣2,1+𝑣1,0̇𝑣2,2𝑝2+𝑒𝑣22,1+𝑣2,1̇𝑣2,2+𝑣2,2̇𝑣2,1+𝑣2,0̇𝑣2,3+𝑣1,1̇𝑣2,2+𝑣1,2̇𝑣2,1+𝑣1,0̇𝑣2,3+𝑒𝑣1,0𝑣2,2+𝑒𝑣1,1𝑣2,1𝑐𝑣1,0𝑣2,2𝑐𝑣1,1𝑣2,1+𝑒𝑣1,2𝑣2,0𝑐𝑣1,2𝑣2,0+2𝑒𝑣2,0𝑣2,2𝑝3+=0.(2.5) In order to obtain the unknown of 𝜈𝑖,𝑗(𝑥,𝑡),𝑖,𝑗=1,2,3,, we must construct and solve the following system which includes 6 equations, considering the initial conditions of 𝜈𝑖,𝑗(0)=0,𝑖,𝑗=1,2,3, :𝑣2,0̇𝑣1,0+𝑣1,0̇𝑣1,0𝑣=0,31,0𝑣21,0+𝑣1,0̇𝑣1,1+𝑣2,0̇𝑣1,1+𝑣1,0𝑣2,0+𝑏𝑣1,0𝑣2,0𝑣1,0𝑣2,0+𝑣2,0𝑣21,0+𝑟𝑣21,0𝑣=0,1,1̇𝑣1,1+𝑣1,0̇𝑣1,2+𝑣2,0̇𝑣1,2+𝑣2,1̇𝑣1,1+2𝑟𝑣1,0𝑣1,1+𝑏𝑣1,0𝑣2,1+2𝑣2,0𝑣1,0𝑣1,1+𝑟𝑣1,1𝑣2,0+𝑟𝑣1,0𝑣2,1+𝑏𝑣1,1𝑣2,0𝑣1,0𝑣2,1𝑣1,1𝑣2,0+𝑣2,1𝑣21,02𝑣1,0𝑣1,1+3𝑣21,0𝑣1,1𝑣=0,2,0̇𝑣2,0+𝑣1,0̇𝑣2,0=0,𝑒𝑣1,0𝑣2,0𝑐𝑣1,0𝑣2,0+𝑣2,0̇𝑣2,1+𝑣1,0̇𝑣2,1+𝑒𝑣22,0𝑣=0,2,1̇𝑣2,1+𝑒𝑣1,0𝑣2,1𝑐𝑣1,0𝑣2,1+𝑒𝑣1,1𝑣2,0𝑐𝑣1,1𝑣2,0+2𝑒𝑣2,0𝑣2,1+𝑣2,0̇𝑣2,2+𝑣1,1̇𝑣2,1+𝑣1,0̇𝑣2,2=0.(2.6) From (2.4), if the first three approximations are sufficient, then setting 𝑝=1 yields the approximate solution of (1.1) to𝑥(𝑡)=lim𝑝1𝑣1(𝑡)=𝑘=3𝑘=0𝑣1,𝑘(𝑡),𝑦(𝑡)=lim𝑝1𝑣2(𝑡)=𝑘=3𝑘=0𝑣2,𝑘(𝑡).(2.7) Therefore,v1,0(𝑡)=𝑥0𝑣(𝑡)=𝑥(0),(2.8)1,1x(𝑡)=0𝑥20𝑥0𝑦0+𝑥0𝑦0+𝑟𝑦0+𝑏𝑦0+𝑟𝑥0𝑡𝑥0+𝑦0,𝑣(2.9)1,21(𝑡)=2𝑥0+𝑦03𝑥0𝑡23𝑦0𝑥20𝑥20𝑏𝑦0+2𝑥30𝑏𝑦0+3𝑥40𝑟+6𝑥30𝑦203𝑦30𝑥0+𝑥30𝑟29𝑥30𝑦0+6𝑥40𝑦09𝑥20𝑦20+2𝑦30𝑥202𝑥30𝑟2𝑟𝑦302𝑏𝑦30+𝑏2𝑦30+𝑟2𝑦30+𝑥20𝑏𝑦0𝑟+3𝑥0𝑟𝑦20𝑏+𝑦20𝑥0𝑒𝑏+𝑏𝑥20𝑦0𝑒𝑏𝑥20𝑦0𝑐3𝑥0𝑏𝑦20+3𝑥0𝑦20+3𝑦30𝑥0𝑟+3𝑦30𝑥0𝑏6𝑥20𝑟𝑦0+2𝑥503𝑥40+𝑦30+2𝑟𝑦30𝑏+9𝑥30𝑟𝑦06𝑥0𝑟𝑦20+9𝑥20𝑦20𝑟+5𝑥20𝑦20𝑏+𝑥30+3𝑥0𝑟2𝑦20+3𝑥20𝑟2𝑦0,𝑣(2.10)2,0(𝑡)=𝑦0𝑣(𝑡)=𝑦(0),(2.11)2,1y(𝑡)=0𝑒𝑥0+𝑐𝑥0𝑒𝑦0𝑡𝑦0+𝑥0,𝑣(2.12)2,21(𝑡)=2𝑦0+𝑥03𝑦0𝑡23𝑦0𝑒𝑥20𝑐+𝑦20𝑐𝑥0𝑒+2𝑒𝑥30𝑐𝑐𝑥20𝑦0𝑐𝑥0𝑦20𝑐2𝑥30+𝑐𝑥30𝑦0+𝑐𝑥20𝑦0𝑟+𝑐𝑥0𝑦20𝑏+𝑐𝑥20𝑦20+𝑐𝑥0𝑦20𝑟𝑒2𝑥303𝑦0𝑒2𝑥203𝑦20𝑒2𝑥0𝑦30𝑒2.(2.13) We also obtained 𝑣1,3 and 𝑣2,3, but because they were too long to maintain, we skip them and only use them in the final numerical results. In this manner, the other components can be easily obtained by substituting (2.8) through (2.13) into (2.7) as follows:x𝑥(𝑡)=𝑥(0)0𝑥20𝑥0𝑦0+𝑥0𝑦0+𝑟𝑦0+𝑏𝑦0+𝑟𝑥0𝑡𝑥0+𝑦0+12𝑥0+𝑦03𝑥0𝑡2(3𝑦0𝑥20𝑥20𝑏𝑦0+2𝑥30𝑏𝑦0+3𝑥40𝑟+6𝑥30𝑦203𝑦30𝑥0+𝑥30𝑟29𝑥30𝑦0+6𝑥40𝑦09𝑥20𝑦20+2𝑦30𝑥202𝑥30𝑟2𝑟𝑦302𝑏𝑦30+𝑏2𝑦30+𝑟2𝑦30+𝑥20𝑏𝑦0𝑟+3𝑥0𝑟𝑦20𝑏+𝑦20𝑥0𝑒𝑏+𝑏𝑥20𝑦0𝑒𝑏𝑥20𝑦0𝑐3𝑥0𝑏𝑦20+3𝑥0𝑦20+3𝑦30𝑥0𝑟+3𝑦30𝑥0𝑏6𝑥20𝑟𝑦0+2𝑥503𝑥40+𝑦30+2𝑟𝑦30𝑏+9𝑥30𝑟𝑦06𝑥0𝑟𝑦20+9𝑥20𝑦20𝑟+5𝑥20𝑦20𝑏+𝑥30+3𝑥0𝑟2𝑦20+3𝑥20𝑟2𝑦0+𝑣1,3y,𝑦(𝑡)=𝑦(0)+0𝑒𝑥0+𝑐𝑥0𝑒𝑦0𝑡𝑦0+𝑥012𝑦0+𝑥03×𝑦0𝑡23𝑦0𝑒𝑥20𝑐+𝑦20𝑐𝑥0𝑒+2𝑒𝑥30𝑐𝑐𝑥20𝑦0𝑐𝑥0𝑦20𝑐2𝑥30+𝑐𝑥30𝑦0+𝑐𝑥20𝑦0𝑟+𝑐𝑥0𝑦20𝑏+𝑐𝑥20𝑦20+𝑐𝑥0𝑦20𝑟𝑒2𝑥303𝑦0𝑒2𝑥203𝑦20𝑒2𝑥0𝑦30𝑒2+𝑣2,3.(2.14)

3. Numerical Results and Comparison with ADM

For comparison with the results obtained by ADM [18], the parameter values in four cases are considered in Table 1.

tab1
Table 1: Parameter values used for illustration purposes.

Results of four terms approximation for 𝑥(𝑡),𝑦(𝑡) obtained by using HPM and ADM [18] are presented in (3.1), respectively: Case1𝑥0.50.35𝑡+0.19476𝑡20.107288𝑡3,𝑦0.30.1125𝑡+0.018808𝑡20.0011284𝑡3,Case2𝑥0.5+0.05𝑡+0.012265𝑡20.0016032𝑡3,𝑦0.30.1125𝑡+0.024433𝑡20.00398199𝑡3,Case3𝑥0.3+0.0799t+0.00533t20.00115𝑡3,𝑦0.60.08𝑡+0.01866𝑡20.00231𝑡3,Case4𝑥0.5+0.07857𝑡0.016020𝑡20.00119873𝑡3,𝑦0.2+0.051428𝑡+0.0055918𝑡2+0.00002245𝑡3,Case1𝑥0.50.35000𝑡+0.19476𝑡20.10728𝑡3,𝑦0.30.11250𝑡+0.018809𝑡20.0011286𝑡3,Case2𝑥0.5+0.05000𝑡+0.012266𝑡20.0016034𝑡3,𝑦0.30.11250𝑡+0.024434𝑡20.0039821𝑡3,Case3𝑥0.3+0.08000t+0.005333t20.0011555𝑡3,𝑦0.60.08000𝑡+0.018667𝑡20.0023112𝑡3,Case4𝑥0.5+0.07857𝑡0.016021𝑡20.0011984𝑡3,𝑦0.2+0.051430𝑡+0.0055920𝑡2+0.00002246𝑡3.(3.1) Figures 14 show the relations between prey and predator populations versus time.

945420.fig.001
Figure 1: Population fraction versus time for Case 1: 𝑟=0.9: (—) prey population fraction; () predator population fraction.
945420.fig.002
Figure 2: Population fraction versus time for Case 2: 𝑟=0.1: (—) prey population fraction; () predator population fraction.
945420.fig.003
Figure 3: Population fraction versus time for Case 3: 𝑟=0.1: (—) prey population fraction; () predator population fraction.
945420.fig.004
Figure 4: Population fraction versus time for Case 4: 𝑟=0.2: (—) prey population fraction; () predator population fraction.

A noteworthy observation from Figure 1 is that prey and predator species can become extinct simultaneously for some values of parameters, regardless of the initial values. Thus, overexploitation of the prey population by constant effort harvesting process together with high predator capturing rate may lead to mutual extinction as a possible outcome of predator-pray interaction. In Figure 2, only the predator population gradually decreases and becomes extinct despite the availability of increasing prey population. This can be attributed to the effect of the predator death rate, being greater than the conversion rate and low constant prey harvesting as shown in Case 2 (see Table 1). Figures 3 and 4 illustrate the possibility of predator and prey long-term coexistence. Depending on the initial values, both prey and predator populations increase or reduce in order to allow long-term coexistence [18].

4. Conclusion

Homotopy perturbation method was employed to approximate the solution of the ratio-dependent predator-prey system with constant effort prey harvesting. The results obtained here were compared with results of Adomian decomposition method. The results show that there is less computations needed in comparison to ADM.

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