Hopf Point Analysis for Ratio-Dependent Food Chain Models

In this paper periodic and quasi-periodic behavior of a food chain model with three trophic levels are studied. Michaelis-Menten type ratio-dependent functional response is considered. There are two equilibrium points of the system. It is found out that at most one of these equilibrium points is stable at a time. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. However for the second equilibrium point the computation is more tedious and bifurcation points can only be found by numerical experiments. It has been found that around these points there are periodic solutions and when this point is unstable, the solution is an enlarging spiral from inside and approaches to a limit cycle.


INTRODUCTION
The term "ratio-dependent predation" is introduced in [1] to describe situations in which the feeding rates of predators depend on the ratio of the number of preys to the number of predators rather than on prey density alone, as is the case in most classical models. One advantage of the ratio dependence is that they prevent paradoxes of enrichment and biological control predicted by classical models [2,3].
Experimental observations [4] suggest that prey dependent models are appropriate in homogeneous situations and ratio-dependent models are good in heterogeneous cases. By many investigators [4,5] it has also been concluded that natural systems are closer to the models with ratio dependence than to the ones with prey density dependence [6].
Generally, a ratio-dependent predator-prey model leads to a system of nonlinear ordinary differential equations. The classical food chain models with only two trophic levels are shown to be insufficient to produce realistic dynamics [12][13][14][15][16]. Therefore we consider the following three trophic levels food chain model with ratio-dependence which is a simple relation between the populations of the three species: z prey on y and only y, and y prey on x and nutrient recycling is not accounted for. After non dimensionalization we have the following system: Where x, y, z stand for the non dimensional population density of the prey, predator and top predator respectively. For , , , η are the yield constants, maximal predator growth rates, half-saturation constants and predators' death rates, r is the prey intrinsic growth rate.

EQUILIBRIUM POINTS
Equilibrium points are the solutions of the nonlinear algebraic system of equations [17,18], Considering the nonnegative ness of the parameters and unknowns, we get two equilibrium points. One of them is of the form is a nonnegative equilibrium point of the system (1) if ( ) . and 0 1 is an interior equilibrium of the system (1) if

STABILITY OF EQUILIBRIUMS
The dynamical behavior of equilibrium points is studied by computation of the eigenvalues of the variational matrix J ;    does not lie in the physical space. Hence the system can not have two stable equilibrium points for the same set of parameters.

For the equilibrium point E 1 :
It can be shown that the real parts of the roots of the cubic algebraic equation Using equilibrium conditions obtained from (2), we see that It can be shown that the coefficients Therefore the characteristic equation of the Jacobi matrix (12) has roots with all negative real parts, and hence 1 E is a stable equilibrium point under these conditions [21].

HOPF BIFURCATION POINTS
When interested in periodic or quasi periodic behavior of a dynamical system, Hopf points are the points which are first to be considered. If we write the autonomous system (1) in the form where ) , , , We say that an ordered pair has two complex conjugate eigenvalues around Inequalities in (4) that guaranties the existence of The last condition  10 c calculated from (20) ;   530371  0  3  092  24  1  0695709  0  2  24092  1  0695709  0 For an initial point close to 1 E , the solution is an enlarging spiral as seen in Fig.6 , the solution enlarges by the time, and eventually reaches to a limit cycle with period 8.37 as shown in Fig. 7. ., 1 1 = µ , the solution has a limit cycle with period 8.37.

VANISHING TOP PREDATOR
When the top predator vanishes, some nonlinear oscillatory phenomena for the first predator and pray occurs. Consider the system which is obtained deleting z from (1).
Considering the nonnegative ness of the parameters and unknowns, we get two equilibrium points. One of them is For the equilibrium point    , the solution is periodic with period 31.83 as seen in Fig.8. , the solution is a spiral as seen in Fig.9. , the solution is an enlarging spiral as seen in Fig.10.

CONCLUSION
In this study, a ratio-dependent food chain model is analyzed and possible dynamical behavior of this system investigated at equilibrium points. It has been shown that, the solutions posses Hopf bifurcations. The system has periodic solutions in a small neighborhood of centers. For unstable nodes, for suitable initial conditions, it is seen that the system undergoes limit cycles. In the case of vanishing top predator, the equilibrium points, the stability of solutions, the existence of limit cycles, the Hopf bifurcation and stability of the periodic solution created by the bifurcation are all studied. A limit cycle in the two dimensional stable manifold is seen to be a periodic solution in the three dimensional system.