On the Dynamics of a General Predator-Prey System

Problem statement: In this study a general two dimensional predator-p rey model is considered. The dynamic and existence of equilibriu m points are studied. Conclusion/Recommendations: Hopf bifurcation is discussed. The existence and u iqueness of limit cycles is proved. Special cases are considered to j ustify our results.


INTRODUCTION
There has been a demanding need for developing and analyzing models of interacting species in ecosystems. Predator-prey models are one of the most important models of two-species interaction. In this study we are concerned with a general 2-dimensional predator-prey system of the form where, x and y are the prey and the predator population sizes, respectively. The parameters s, D and k are positive and represent the conversion efficiency rate of prey to predator, predator death rate and the carrying capacity of the prey population, respectively. The parameters E 1 ≥ 0, E 2 ≥ 0 denote the harvesting efforts for the predator and prey respectively. The expressions q 1 E 1 x and q 2 E 2 x represent the catch of the respective species, where q 1 and q 2 denote the catch ability coefficients of the prey and predator, respectively. The functions g(x) and p(x) denote the functional response of the prey and predator, respectively and satisfy the assumptions p(0) = 0 and g(0)>0, for all x>0. There have been considerable interests in the dynamics of the predatorprey models of some special cases of (1) by several authors (Attili, 2001;Attili and Mallak, 2006;Xiao and Ruan, 2001;Hesaaraki and Moghadas, 2001;Kar and Matsuda, 2007;Moghadas and Corbett, 2008;Moghadas and Alexander, 2005;Sugie et al., 1997;Ruan and Xiao, 2001). Hasik (2000); Sesay et al. (2010); Moghadas and Alexander (2006); Saha and Bandyopadhyay (2005) and Tao et al. (2011) the authors considered Eq. 1 in the special The aim of this study is to discuss the qualitative properties of the general predator-prey system (1). We discuss the existence and stability of equilibria and nonexistence criteria for limit cycles. We explore the uniqueness of limit cycles using Kuang and Freedmann approach (Moghadas and Corbett, 2008) and some applications. It is natural due to biological considerations to expect that the solutions of (1) must to be positive and bounded. So we give the following result which is a partial extension of those of (Freedman and So, 1985) and (Saha and Bandyopadhyay, 2005).The paper end with a brief discussion. Proof: According to the ecological consideration the positivity of the solutions of (1) is obtained. To show the roundedness', we set: where, α = max g(x) and x≤M =(α+r-q 1 E 1 ). Thus applying the theory of differential inequality (Freedman and So, 1985), we obtain: So the limits as tends to ∞ yields Thus we have that all solutions of (1) start in 2 It is clear that y* increases with s and decrease with D. This is a natural conclusion because an increase in the predator death rate will cause decrease in the predator population and hence enhances the survival rate of prey. It is also clear that the positive equilibrium P 2 (x * , y * ) exists for (1) only if the harvesting satisfies The nature of Equilibria: We discuss the stability properties of the equilibria p 0 (0,0), p 1 (x 1 , 0) and p 2 (x * ,y * ) .The Jacobian matrix of (1) around P 0 (0,0) is: i.e., we have two eigenvalues λ 1 =-D-q 2 E 2 and λ 2 =g(0)-q 1 E 1 .

Proof:
Since for any real function α(x, y) we have: choosing Dulac function α=1, then: By Bendixon-Dulac Theorem (Kelley and Peterson, 2003), this is sufficient for nonexistence of periodic solutions of (1). The following result discusses the existence of limit cycles.
Theorem 3.1: The system (1) has at least one limit cycle in Ω = {(x,y):x >0, y>0} if and only if Eq. 2: In order to prove sufficient condition of Theorem 3.1, we first note that one can show ( (Freedman and So, 1985;Moghadas and Corbett, 2008) that, if: where, M 2 =max x then: • The set ϑ is positively invariant x , y , x t , y t as t + ∈ →ϑ → ∞ ℝ Now since by above, the characteristic polynomial at the nontrivial positive equilibrium P 2 (x * , y * ) is: Since sy * p'(x * ) p (x * )>0, the roots of p(λ) have positive real parts if and only if (2) holds. Cosequently P 2 (x * , y * ) is unstable if (2) holds. It is easy to check that the stable and unstable manifolds at (0, 0) are on the xaxis and y-axis, respectively. If P 2 (x * ,y * ) is unstable, then by above discussion and Poincaré Bendixson Theorem, it follows that the ω -limit set of every solution initiating at a point in the first quadrant is a limit cycle. Therefore, we have established that if (2) holds, then the system (1) has at least one limit cycle. This proves the sufficient condition.

Remark 3.1:
We note that it can be shown that this condition is not only sufficient but also necessary (Attili, 2001;Attili and Mallak, 2006;Hesaaraki and Moghadas, 2001;Moghadas and Corbett, 2008).

Uniqueness of limit cycles:
We discuss the uniqueness of limit cycles of the general system (1). Our criterion improves and partially generalizes those of (Hasik, 2000;Kuang and Freedman, 1988;Sugie et al., 1997). We first start with the most famous uniqueness result which was the motivation for several authors and criteria. As in (Kuang and Freedman, 1988) we consider the system Eq. 3: where, γ>0, all the functions are sufficiently smooth on [0,∞) and satisfy Eq. 4: The authors in (Kuang and Freedman, 1988) gave the following result on the uniqueness of limit cycles.
Theorem 4.1: Kuang and Freedman (1988) if there exist constants x * and m with 0< x * < m such that: Then the system (3) has exactly one limit cycle which is globally asymptotically stable. In view of Theorem 4.1 and those of (Sugie et al., 1997;Hasik, 2000), we give the following uniqueness theorem for the limit cycles of our general system (1).
Therefore the condition of Theorem 4.1 (Freedman and So, 1985) is satisfied and this completes the uniqueness of limit cycles.
Applications: Now we give some examples from the ecological literature. The numerical simulations may justify the results.
It is clear that . Thus the condition of Theorem 3.1. holds. Then there exists at least one limit cycle (Fig. 1).
It is clear that . Thus the condition of Theorem 3.1. Holds.
Then there exists at least one limit cycles (Fig. 2).
It is clear that Then there exists at least one limit cycles (Fig. 3).
It is clear that Then there exists at least one limit cycle (Fig. 4). Remark 5.2: we may note that our examples and graphs are different from those exist in (Kar and Matsuda, 2007;Moghadas and Corbett, 2008).

COUCLUSION
In this article we discuss the existence and stability of equilibrium points using Routh-Hurwitz approach. Theorem1 shows that all solutions of the model are positive and bounded. We give sufficient condition guarantees that the model has at least one limit cycle in the first quadrant of the xy plane. We give sufficient condition guarantees that the model has at least one limit cycle in the first quadrant of the xy plane. We give sufficient condition guarantees that the model has at least one limit cycle in the first quadrant of the xy plane if: 1 1 x *g '(x*) g(x*) y * p '(x*) _ q E 0 + − < Using Bendixon Dulac Theorem we discuss the case for which the predator-prey model (1) has no periodic solution. Using Kuang and Freedman technique we give sufficient condition for the uniqueness of limit cycles of (1). Numerical examples for some special cases of g(x) are given to justify the results with graphs showing the existence of limit cycles.