Spirals and Cycles of Biological Systems via Extended Rosenzweig-MacArthur Model with Ratio-dependent Functional Response

This paper investigates stable proper nodes, stable spiral sinks and stable ω−limit cycles of Extended RosenzweigMacAthur Model, which incorporates ratio-dependent functional response on predation mechanism. The ultimate boundedness condition has been used to predict extinction, co-existence, and exponential convergence scenarios of the model. The Poincare-Bendixson results guarantee existence of periodic cycles of the models. The system degenerate from stable spiral sinks to stable ω-limit cycles as control parameter varies. Numerical simulations are provided to support the validity of theoretical findings.


Introduction
The theory of nonlinear dynamical systems has been robustly explored in explaining, interpreting, and predicting the qualitative behaviors of ecological populations of interacting species.Rosenzweig and MacArthur (1963) formulated and studied the qualitative behaviors of a di-trophic food chain model, with Holling type II functional response given as; b 1 +x 1 (t) x 2 (t) b 1 +x 1 (t) x 2 (t) − d 2 x 2 (t) (1) which shows stability behaviors of predator-prey populations.This two-dimensional model exhibits a unique global attractor which is either an equilibrium or limit cycle as well as dynamical behaviors with deviated arguments (Wrosek, 1990, Shi, 2013).In three-dimensional systems, a tri-trophic food chain model as an extension of the Rosenzweig-MacArthur predator-prey model was formulated and studied; x 2 (t) x 3 (t) x 3 (t) − d 3 x 3 (t) (2) Qualitative behaviors such as stability of equilibrium points, local and global bifurcations, limit cycles, peak-topeak dynamics on this model with constant or seasonal varying of parameters has been investigated in system (2) (Feo, & Rinaldi, 1997;Kutnetsov, & Rinaldi, 1996;Kutnestsov, Rinaldi, 2001;Candaten, & Rinaldi, 2003).Feng, Rocco, Freeze, and Lu (2014), modified the Rosenzweig-MacArthur model in three dimensional for a more general and complex, but realistic model as; which exhibits rich dynamical complexity of ecological population species.A topological equivalence dynamical system of system (3) were formulated via non-dimensionalization of the state variables as follows; where  Joshua, & Akpan, 2018).In this article, system (4)is modified with a ratio-dependent functional response and its dynamical complexity is studied.

Model Description and Existence of Bounded Solutions
Consider the ratio-dependent functional response incorporated in the predation mechanism of Extended Rosenzweig-MacArthur Model as follows subject to initial conditions (x(0) = x 0 , y(0) = y 0 , z(0) = z 0 ), where x(t), y(t), z(t) are the populations of interacting species; preys, predators and super-predators respectively.The ecological parameters; α is the preys growth rate, κ is the environmental carrying capacity of the prey, η is the maximum predation rate on prey, ε is the maximum biomass conversion efficiency of the predator, ξ is the natural death rate of the predator, β is the maximum superpredator biomass conversion efficiency, and µ is the natural death rate of the super-predators.The population density function of system (1) is continuously differentiable in the non-negative cone of the state space R 3 + = (x(t), y(t), z(t)|x(t) ≥ 0, y(t) ≥ 0, z(t) ≥ 0) ∀t ≥ 0. Assume that the long-term survival of super-predators is dependent on the prey, predator abundance as well as its biomass conversion efficiency, then the steady state behavior and co-existence fixed point E * (x * , y * , z * ) of system (5) satisfies the equations; Using differential inequalities and standard comparison argument, the phase flows φ t (t 0 ; x(t), y(t), z(t)) of system (5) and ( 6) are ultimately bounded in the compact invariant region Ω ∈ Int(R 3 + ) ∀t ≥ 0 defined as Proposition 1 (Agarwal,O'Regan, & Saker, 2014): Given that Ω is a closed, convex, and nonempty subset of a Banach space R 3 + .Denote the system (5) as a vector differentials; Ẋ(t) = F(X(t), ρ * ); where ρ * is a control parameter, X(t) = ((x(t), y(t), z(t)) T and F : R 3 + × R + → R 3 + is a continuous mapping with F(Ω) a relatively compact subset of R 3 + .Then F has at least one fixed point(equilibrium point), and Ω is a global attractor of every phase flows φ t (t 0 ; x(t), y(t), z(t)) of system (5), if α > η + 1, ε + σ > ξ, β > 0.
Remarks The proposition established the extinction, co-existence and exponential convergence scenarios of interacting population species of system (5) Definition 1 (Perko, 2001): A limit cycle γ of the dynamical system (5) in the plane is a periodic orbit which is a α or ω− limit set of a trajectory γ other than γ.If a limit cycle γ is the ω − limit set of every trajectory in a neighborhood of γ, γ is said to be an ω − limit cycle or stable limit cycle.Likewise, if γ is the α − limit set of neighboring trajectories of γ, γ is said to be an α − limit cycle or unstable limit cycle Definition 2 (Wiggins, 2003): • maps orbits of the first system in U onto orbits of the second system in V = F(U) ⊂ R 3 + , preserving the direction of time.
Remarks Definition (2) provides us with a way of characterizing two vector fields of the same qualitative dynamics.
Definition 3 (Poincare-Bendixson conditions: Brauer & Castillo-Chavez, 2012 ) Given that Ω is a positively invariant region for the vector field function F containing a finite number of fixed points.Let p ∈ Ω, and consider ω(p).Then one of the following possibilities holds; • ω(p) is a fixed point.

Equilibrium Points and Linearized Jacobian of Planar Subsystem
We obtain the steady state equilibrium points of system (6), by solving the planar sub-systems independent of time, and deduce the positivity conditions.The model exhibits the following trivial and semi-trivial equilibrium points.
Thus, the linearized Jacobian of system (5) in the neighborhood of any equilibrium point E(x * , y * , z * ) yields,
Proposition 3: The population of prey-predator species in the neighborhood of the equilibrium point E 2 exhibits stable spiral sink, if A 1 A 2 − A 3 > 0 and eigenvalues (18) λ i (i = 1, 2, 3) have negative real parts, otherwise unstable.It degenerates to a stable ω−limit cycle if A 1 A 2 − A 3 = 0, eigenvalues λ i (i = 1, 2, 3) have purely imaginary parts.and Similarly, the Jacobian matrix (9) evaluated at the prey super-predator equilibrium point E 3 yields; Its satisfies the characteristic polynomial, where with eigenvalues, λ i (i = 1, 2, 3) defined as follows: By Routh-Hurwitz conditions and Descartes rule of sign, the next proposition follows.
Proposition 4: The population of prey super-predator species in the neighborhood of the equilibrium point E 3 exhibits stable spiral sink, if A 1 A 2 − A 3 > 0 and eigenvalues (22) λ i (i = 1, 2, 3) have negative real parts, otherwise unstable.It degenerates to a stable ω−limit cycle if A 1 A 2 − A 3 = 0 and eigenvalues (22) λ i (i = 1, 2, 3) have purely imaginary parts.

Dynamical Behaviors of Positive Coexistence Equilibrium Point
System (10) has a unique positive coexisting equilibrium point say, E 4 (u * , y * , v * ) defined as follows; Also, the Jacobian matrix J i j (i, j = 1, 2, 3) of system (10) evaluated at the coexistence equilibrium point and the corresponding characteristic polynomial yields, Using Routh-Hurwitz conditions (see, Wiggins, 2003), the dynamical behaviors of system (5) at coexisting equilibrium point is established in the following proposition.
Proposition 5: The coexistence equilibrium point E 4 (u * , y * , v * ) for the model ( 5) or ( 10) is asymptotically stable if and only if the ecological parameters satisfies the condition (25), and degenerates to a stable ω − limit cycle for some ecological parameters, if condition (26) holds.

Conclusion
The study investigated qualitative dynamical behaviors of an extended Rosenzweig-MacArthur model with ratiodependent functional response on predation mechanism.Using the theory of nonlinear dynamical systems, we established existence and boundedness of solutions in real parameter space.Some pseudo-codes in maple 16 on dynamical systems were used to reduce the algebraic complexity of the model.The result of numerical simulations were plotted as phase portraits, and phase space diagrams to verify the propositions established.The model shown stable spirals, stable proper nodes and stable limit cycles via appropriate variations of the ecological parameters.Thus, incorporating ratio-dependent functional response unfolded a robust and realistic dynamics of the models.

Figure 1 .
Figure 1.Phase space diagram at origin

Figure 2 .
Figure 2. Phase Portrait of Prey Population