Hopf-bifurcation Limit Cycles of an Extended Rosenzweig-MacArthur Model

In this paper, we formulated a new topologically equivalence dynamics of an Extended Rosenzweig-MacArthur Model. Also, we investigated the local stability criteria, and determine the existence of co-dimension-1 Hopf-bifurcation limit cycles as the bifurcation-parameter changes. We discussed the dynamical complexities of this model using numerical responses, solution curves and phase-space diagrams.


Introduction
In general, any nonlinear dynamical system contains certain parameters called bifurcation parameters or controlled-free parameters, and thus it's an imperative to study the qualitative behaviors of such robust systems as the parameters are varied.This study of bifurcation analysis includes the post-critical behaviors of the nonlinear system in the neighborhood of the critical points called Hopf-bifurcation limit cycles or periodic solutions (Liao &Yu, 2007).The complexities of such nonlinear dynamics include heteroclinic orbits, homoclinic orbits and chaos which envisaged essential global behaviors (Kutnzetsov, 1995;Sun & Luo 2005;Wang & Zhao, 2011).

Model Formulation and Boundedness
The Extended Rosenzweig-MacArthur (ERM) model formulated and studied by Feng, Freeze, Lu, and Rocco (2014) is given as: where,  1 (),  2 (),   3 () are population biomass densities of prey, predator and super-predator respectively, and constant parameters with significance ecological implications.

Existence and Positivity of Trivial and Semi-trivial Equilibria of the Model
We obtain the critical point of model (1.2), by solving the system during it steady-state; independent of time, and deduce the positivity conditions of each critical point.The model exhibited the following trivial, and semi-trivial equilibria points: Descartes' rule of sign variation guarantees the existence of coexistence equilibrium of the model as stated in the following lemma:

Lemma III: Stability of Prey-Predator Equilibrium Point.
The equilibrium point Locally asymptotically stable if Consider the community matric of the coexistence equilibrium point,  4 ( =  * ,  =  * ,  =  * ); where, The characteristic polynomial of the community matrix yields Then, applying Routh-Hurwitz conditions on the characteristic polynomial yields, where  1 ,  2 ,   3 are coefficient of characteristic polynomial of the community matrix at the coexistence equilibrium point.

Existence of Hopf-bifurcation at
Proof: Evaluate the community matrix (4.0) at the equilibrium point, Consider the characteristic polynomial of prey-predator community matrix, say; solving quadratic expression yields the roots as; Hopf-bifurcation occurs when,  1,2 = ± 2 () , solving  1 () = 0, yields the critical bifurcating threshold say; Suppose the positive invariant parameter space ℜ + 4 (, , , ) is a smooth analytic function of bifurcation parameter, w.r.t  , the transversality condition follows as; Hence, the prove is complete.is asymptotically stable for  ∈ (0,  * ), and degenerates to a stable limit cycle for  >  * .Then, model (3.0) exhibit a supercritical Hopf-bifurcation at the equilibrium if,
Then the model (1.2) degenerates to a family of periodic cycles, bifurcating the equilibrium point,  4 ( =  * ,  =  * ,  =  * ) in the neighborhood of the critical bifurcating threshold  * such that  ∈ ( * − ,  * + );  > 0 provided the following conditions are satisfied, Proof: Consider the characteristic polynomial of the coexistence equilibrium point and assume the roots are non-zero, as  1 ≠ 0. Set the eigenvalues as smooth analytic function of the bifurcation parameter,  as; Then, at critical bifurcating threshold,  =  * we have that the pair of complex conjugate degenerates, and satisfies On substituting eqn.(1.7) into the characteristic polynomial of coexistence equilibrium point, equating real and imaginary parts to zero yields; Hence the critical bifurcating threshold occurs on a smooth analytic function, say; Next, it suffices to verify the transversality condition, substitute  1 =  1 + 1 into the characteristic polynomial, and equate real and imaginary parts yields; Hence, there exist a family of periodic solutions bifurcating from positive coexistence equilibrium point  4 ( =  * ,  =  * ,  =  * ) in the neighborhood of the bifurcating threshold,  =  * .That is Hopf-bifurcation occurs when the bifurcation parameter is open for  ∈ ( * − ,  * + ).

Hopf-bifurcation Limit Cycle of Prey-predator Equilibrium Point
Given the bifurcation parameter , observe that at critical bifurcating threshold  * = 4.3333, a stable limit cycle

Conclusion
In this paper, we study a new dimensionless model as an extension of the Rosenzweig-MacArthur model; a tri-trophic food chain model that depicts and predicts the populations of interacting species in the natural ecological system.We established ultimate boundedness conditions and exponential convergence of populations of the interacting species.
Using local stability conditions, we determine the existence of codimension-1 Hopf-bifurcation limit cycles of the model as the free-parameter changes.Ecologically, as the super-predator biomass efficiency increases through its ecological interactions with other species, the system oscillates periodically, and the species coexisted based on qualitative theory of dynamical system.