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

Authors: Abdoul R. Ghotbi; A. Barari; D. D. Ganji
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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 parameterp∈[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 [1–7] and was successfully applied to
various engineering problems.The motivation of this paper is
to extend the homotopy perturbation method (HPM) [8–17] 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 withr,
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 as(1.1)dxdt=x(1−x)−bxyy+x−rx,dydt=cxyy+x−ey,  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,(2.1)H(ν,p)=(1−p)[L(ν)−L(u0)]+p[A(ν)−f(r)]=0,p∈[0,1],rεΩ,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: (2.2)(1−p)(ν2ν˙1+ν1ν˙1−x˙0y0−x˙0x0)+p(ν2ν˙1+ν1ν˙1−(1−b−r)ν1ν2+ν2ν12−(1−r)ν12+ν13)=0,(1−p)×(ν2ν˙2+ν1v˙2−y˙0y0−x0y˙0)+p(ν2ν˙2+ν1ν˙2+(e−c)ν1ν2+eν22)=0,
where dot denotes differentiation with respect to t, and the
initial approximations are as follows:(2.3)v1,0(t)=x0(t)=x(0),v2,0(t)=y0(t)=y(0). Assume that the solution of (2.2)
can be written as a power series in p as follows:(2.4)ν1=ν1,0+pν1,1+p2ν1,2+p3ν1,3+⋯,ν2=ν2,0+pν2,1+p2ν2,2+p3ν2,3+⋯, 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 have(2.5)(v2,0v˙1,0+v1,0v˙1,0)+(v1,03−v1,02+v1,0v˙1,1+v2,0v˙1,1+rv1,0v2,0+bv1,0v2,0−v1,0v2,0+v2,0v1,02+rv1,02)p+(v1,1v˙1,1+v1,0v˙1,2+v2,0v˙1,2+v2,1v˙1,1+2rv1,0v1,1+bv1,0v2,1+2v2,0v1,0v1,1+rv1,1v2,0+rv1,0v2,1+bv1,1v2,0−v1,0v2,1−v1,1v2,0+v2,1v1,02−2v1,0v1,1+3v1,02v1,1)p2+(v1,1v˙1,2+v1,2v˙1,1+v1,0v˙1,3+v2,1v˙1,2+v2,0v˙1,3+v2,2v˙1,1+v2,0v1,12−v1,0v2,2−v1,2v2,0−v1,1v2,1+v2,2v1,02+rv1,12+3v1,0v1,12−v1,12+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+3v1,02v1,2−2v1,0v1,2)p3+⋯=0,(v2,0v˙2,0+v1,0v˙2,0)+(ev1,0v2,0−cv1,0v2,0+v2,0v˙2,1+v1,0v˙2,1+ev2,02)p+(v2,1v˙2,1+ev1,0v2,1−cv1,0v2,1+ev1,1v2,0−cv1,1v2,0+2ev2,0v2,1+v2,0v˙2,2+v1,1v˙2,1+v1,0v˙2,2)p2+(ev2,12+v2,1v˙2,2+v2,2v˙2,1+v2,0v˙2,3+v1,1v˙2,2+v1,2v˙2,1+v1,0v˙2,3+ev1,0v2,2+ev1,1v2,1−cv1,0v2,2−cv1,1v2,1+ev1,2v2,0−cv1,2v2,0+2ev2,0v2,2)p3+⋯=0.  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,… :(2.6)v2,0v˙1,0+v1,0v˙1,0=0,v1,03−v1,02+v1,0v˙1,1+v2,0v˙1,1+v1,0v2,0+bv1,0v2,0−v1,0v2,0+v2,0v1,02+rv1,02=0,v1,1v˙1,1+v1,0v˙1,2+v2,0v˙1,2+v2,1v˙1,1+2rv1,0v1,1+bv1,0v2,1+2v2,0v1,0v1,1+rv1,1v2,0+rv1,0v2,1+bv1,1v2,0−v1,0v2,1−v1,1v2,0+v2,1v1,02−2v1,0v1,1+3v1,02v1,1=0,v2,0v˙2,0+v1,0v˙2,0=0,ev1,0v2,0−cv1,0v2,0+v2,0v˙2,1+v1,0v˙2,1+ev2,02=0,v2,1v˙2,1+ev1,0v2,1−cv1,0v2,1+ev1,1v2,0−cv1,1v2,0+2ev2,0v2,1+v2,0v˙2,2+v1,1v˙2,1+v1,0v˙2,2=0. From (2.4), if the first three approximations are sufficient, then setting p=1 yields the approximate solution of (1.1)
to(2.7)x(t)=lim⁡p→1v1(t)=∑k=0k=3v1,k(t),y(t)=lim⁡p→1v2(t)=∑k=0k=3v2,k(t). Therefore,(2.8)v1,0(t)=x0(t)=x(0),(2.9)v1,1(t)=−x0(x02−x0−y0+x0y0+ry0+by0+rx0)tx0+y0,(2.10)v1,2(t)=12(x0+y0)3((x0t2(3y0x02−x02by0+2x03by0+3x04r+6x03y02−3y03x0+x03r2−9x03y0+6x04y0−9x02y02+2y03x02−2x03r−2ry03−2by03+b2y03+r2y03+x02by0r+3x0ry02b+y02x0eb+bx02y0e−bx02y0c−3x0by02+3x0y02+3y03x0r+3y03x0b−6x02ry0+2x05−3x04+y03+2ry03b+9x03ry0−6x0ry02+9x02y02r+5x02y02b+x03+3x0r2y02+3x02r2y0))),(2.11)v2,0(t)=y0(t)=y(0),(2.12)v2,1(t)=y0(−ex0+cx0−ey0)ty0+x0,(2.13)v2,2(t)=−12(y0+x0)3((y0t2(3y0ex02c+y02cx0e+2ex03c−cx02y0−cx0y02−c2x03+cx03y0+cx02y0r+cx0y02b+cx02y02+cx0y02r−e2x03−3y0e2x02−3y02e2x0−y03e2))). 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:(2.14)x(t)=x(0)−(x0(x02−x0−y0+x0y0+ry0+by0+rx0)tx0+y0)+12(x0+y0)3(x0t2(3y0x02−x02by0+2x03by0+3x04r+6x03y02−3y03x0+x03r2−9x03y0⁢+6x04y0−9x02y02+2y03x02−2x03r−2ry03−2by03+b2y03+r2y03+x02by0r+3x0ry02b+y02x0eb+bx02y0e−bx02y0c−3x0by02+3x0y02+3y03x0r+3y03x0b−6x02ry0+2x05−3x04+y03+2ry03b+9x03ry0−6x0ry02+9x02y02r+5x02y02b+x03+3x0r2y02+3x02r2y0))+v1,3⋯,y(t)=y(0)+y0(−ex0+cx0−ey0)ty0+x0−12(y0+x0)3×(y0t2(3y0ex02c+y02cx0e+2ex03c−cx02y0−cx0y02−c2x03+cx03y0+cx02y0r+cx0y02b+cx02y02+cx0y02r−e2x03−3y0e2x02−3y02e2x0−y03e2))+v2,3⋯.


## 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.Table 1
Parameter values used for illustration
purposes.
Casex0y0bcer10.50.30.80.20.50.920.50.30.80.20.50.130.50.60.50.50.30.140.50.20.50.50.10.2Results of four terms approximation forx(t),y(t) obtained by using HPM and ADM [18] are
presented in (3.1), respectively:  (3.1)Case1:x≈0.5−0.35t+0.19476t2−0.107288t3,y≈0.3−0.1125t+0.018808t2−0.0011284t3,Case2:x≈0.5+0.05t+0.012265t2−0.0016032t3,y≈0.3−0.1125t+0.024433t2−0.00398199t3,Case3:x≈0.3+0.0799t+0.00533t2−0.00115t3,y≈0.6−0.08t+0.01866t2−0.00231t3,Case4:x≈0.5+0.07857t−0.016020t2−0.00119873t3,y≈0.2+0.051428t+0.0055918t2+0.00002245t3,Case1:x≈0.5−0.35000t+0.19476t2−0.10728t3,y≈0.3−0.11250t+0.018809t2−0.0011286t3,Case2:x≈0.5+0.05000t+0.012266t2−0.0016034t3,y≈0.3−0.11250t+0.024434t2−0.0039821t3,Case3:x≈0.3+0.08000t+0.005333t2−0.0011555t3,y≈0.6−0.08000t+0.018667t2−0.0023112t3,Case4:x≈0.5+0.07857t−0.016021t2−0.0011984t3,y≈0.2+0.051430t+0.0055920t2+0.00002246t3.                   Figures 1–4 show the
relations between prey and predator populations versus time.Figure 1
Population fraction versus time for Case 1:r=0.9: (—) prey population
fraction; (◦◦◦) predator population fraction.Figure 2
Population fraction versus time for Case 2:r=0.1: (—) prey population
fraction; (◦◦◦) predator population
fraction.Figure 3
Population fraction versus time for Case
3:r=0.1: (—) prey population
fraction; (◦◦◦) predator
population fraction.Figure 4
Population fraction versus time for Case
4:r=0.2: (—) prey population
fraction; (◦◦◦) predator
population fraction.A noteworthy observation from Figure1 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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*Source: 945420-2008-09-01.xml*