Global Stability and Hopf-bifurcation Analysis of Biological Systems using Delayed Extended Rosenzweig-MacArthur Model

This paper investigates the global asymptotic stability of a Delayed Extended Rosenzweig-MacArthur Model via Lyapunov-Krasovskii functionals. Frequency sweeping technique ensures stability switches as the delay parameter increases and passes the critical bifurcating threshold.The model exhibits a local Hopf-bifurcation from asymptotically stable oscillatory behaviors to unstable strange chaotic behaviors dependent of the delay parameter values. Hyper-chaotic fluctuations were observed for large delay values far away from the critical delay margin. Numerical simulations of experimental data obtained via non-dimensionalization have shown the applications of theoretical results in ecological population dynamics.


Introduction
In Mathematical modeling of multiple interacting species, time-delays play important role on deterministic qualitative behaviors of dynamical systems.Recent advances in research on delay systems are found in (Gu, Kharitovov & Chen, 2003;Lakshmanan & Senthilkumar, 2010;Smith, 2010;Agarwal, O'Regan, & Saker, 2014;Niculescu, & Gu, 2004).In ecological population dynamical systems, changes in the environmental carrying capacities including maturation and gestation periods may not necessarily affect the entire population dynamics immediately.There are some time-past prior to the present state variables that affect the growth of the interacting species in the long run.These are classified as single or multiple delay parameters, state dependent delays, neutral-time delays, state dependent delays, and distributed delays.Also, the dynamical behaviors of such systems including global stability, instability, Hopf-bifurcations, strange chaotic attractor are induced by delayed parameters.(see, Feng & Hongwei, 2012;Wang & Li 2003;Xu & Wu, 2014;Agrawal, Jana & Upadhyay, 2014;Jatav & Dhar, 2015).Thus, more realistic mathematical models in ecological population dynamics needs to incorporate a time lag or history between the moment in which an action takes place and the moment its effect is actually observed in the dynamical system.There are recent studies in dynamical behaviors of a 3-D Rosenzweig-MacArthur Model (1963) such as uniform boundedness, local and global asymptotic stabilities, local Hopf-bifurcation/limit cycles, persistence and permanence, global attractors, existence and uniqueness of positive periodic solution (see, Feng, Freeze, Lu & Rocco, 2014;Joshua, Akpan, Adebimpe & Madubueze, 2016;Joshua & Akpan 2016;Joshua & Akpan, 2017).However, to the best of the authors' knowledge little emphasis has been placed on the effects of time-delay parameter on the model.Thus, in this paper, the effect of time-delay bifurcation parameters on the dynamical behaviors of the model will be studied.

Existence of Coexisting Equilibrium Point
The system (2) can be represented as Ẋ(t) = F(X(t), X(t − τ)) where X(t) = (x(t), y(t), z(t)) ∈ R 3 + τ ≥ 0 is a discrete delay-in-time parameter and F a vector valued continuous function.Generally, the behaviors of the dynamical system (2) at steady-state are independent of the delay parameter and satisfies F(X(t) = X(t − τ) = X * = 0) and X * = (x * , y * , z * ) ∈ R 3 + being any fixed point of system (2).Observe that, the model depicts an ecological population dynamical system.Thus, the interest will be focused on the behavior of the coexisting equilibrium point E * (x * , y * , z * ) satisfying the following conditions; By applying Descartes' Rule of Sign (Wiggins, 2003), system (2) has a unique positive equilibrium point E * (x * , y * , z * ) if there exists a unique positive real root x * of the polynomial in (3), whence

Global Asymptotic Stability
Definition 1: The time-delay dynamical system (2) is stable independent of delay for all nonnegative delay parameters, otherwise the system's stability is dependent on delay.
Lemma 1 ( Cerone & Dragomir, 2011): If p, q ≥ 1 is defined with conjugacy 1 p + 1 q = 1, then 1 p x p + 1 q y q ≥ xy; ∀x, y ≥ 0 Proposition 1: If the dynamical system (2) is locally asymptotically stable, then the unique fixed point E * (x * , y * , z * ) is globally asymptotically stable independent of delay parameter (in the sense of Lyapunov), given a Lyapunov-Krasovskii functional V(x, y, z) : where and M being a real positive definite and symmetric diagonal matrix.
Proof Consider a candidate Lyapunov-Krasovskii functional given as Now using lemma 1, the directional derivative of the Lyapunov-Krasovskii functional V(x, y, z) along the solution trajectories of system (2) yields; The quadratic bilinear form of equations ( 6) yields; being a positive symmetric matrix defined as; Observe that the Lyapunov-Krasovskii functional satisfies and E(x * , y * , z * ) is a global minimum of V(x, y, z).Thus V(x, y, z) is a strictly positive definite functional.Similarly, it is clear from ( 7) that the symmetric matrix M satisfies the Sylvester's criterion for which all the upper-left leading principal minors of M are positive, and V(x, y, z) < 0. Using Lyapunov-LaSalle invariance principle (LaSalle, 1968), the fixed point E(x * , y * , z * ) is global attractor trapped in the largest invariance subset of R 3 + .Hence every trajectory of system ( 2) is attracted to coexisting equilibrium point E(x * , y * , z * );

Local Hopf-Bifurcation Analysis via Frequency Sweeping Technique
In this section, frequency-domain technique is employed to investigates the distribution of roots of characteristic quasi-polynomial of system (2) in the upper-right half complex plane, critical bifurcating delay margin (τ * > 0), zero-crossing frequency number (ω * > 0) from stable region to unstable region, and transversality condition.However, since only finitely many unstable roots may be in the open right half-plane, there are only finite number of zero crossings.For details on frequency-sweeping technique see (Li, Niculescu & Cela, 2015) Lemma 2: [Ruan & Wei, 2003] Consider the transcendental equation p(λ, e −λτ 1 , ..., e −λτ m ) = λ n + p (0) as (τ 1 , τ 2 , ..., τ m ) vary, the sum of orders of the zeros of p(λ, e −λτ 1 , ..., e −λτ m ) in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.
Remark 1: This means that the number of characteristic roots with positive real parts can change only if there exist a pair of imaginary roots.
Lemma 3 [ Gu, Kharitonov & Ghen, 2003] The time-delay dynamical system (2), defined as; (under appropriate initial conditions, where X(t) ∈ R 3 + is the system steady state at time t, A 0 , A 1 ∈ R 3×3 are real matrices, and τ ∈ [0, +∞) is the delay parameter) is asymptotically stable independent of delay if and only if the following conditions are satisfied; where ρ(.) denotes a spectral radius.For proof of lemma 2 see details in (Gu, Kharitonov & Ghen, 2003) .
Remark 2: When the system is not stable independent of delay, it remains possible to extend the above lemmas to compute the critical delay margin( i.e the first stability interval, including τ = 0) which characterized the delay interval governing asymptotic stability of the system.
Remark 3: The determination of critical pair (τ * , ω * ) is the defining feature of frequency sweeping technique.The results show that, if the gestation delay parameter τ is in the cluster [0, τ * ) the model is stable at the equilibrium point E(x * , y * , z * ).Also, a Hopf-bifurcation occurs at τ = τ * , and the model degenerates to an unstable oscillatory behavior for τ ∈ (τ * , +∞].In this section, a set of experimental data was generated via non-dimensionalization to verify the theoretical results (see table 1).All ecological parameters including the fixed point and initial conditions remain unchanged, and gestation delay parameter will be varied to control the dynamics of the system.

Numerical Simulations
In figure (1) the dynamical system (2) exhibit periodic oscillatory behaviors independent of any delay parameter.The system is locally asymptotically stable at the coexisting equilibrium point for all ecological parameters in table 1 and the Jacobian matrix computed as; yields the negative spectra (-0.09751206712 ± 0.1569067171i, −0.7051110911i).Observe that theorem 1 was satisfied because the symmetric matrix M is positive definite with positive spectra computed using ecological parameters of table 1, . Simulation of global asymptotic stable behaviors of dynamical system (2) at coexisting equilibrium point for delay parameter τ = 0 or τ = +∞.
Thus, the system is globally asymptotically stable in the sense of Lyapunov, and every trajectory converges to the coexisting equilibrium point.Biologically, the populations of the interacting species will persist for a long time without occurrence of extinction scenarios amongst species.