Existence and Uniqueness of Positive Periodic Solution of an Extended Rosenzweig-MacArthur Model via Brouwer’s Topological Degree

The necessary conditions for existence of periodic solutions of an Extended RosenzweigMacArthur model are obtained using Brouwer’s degree. The forward invariant set is formulated to ensure the boundedness of the solutions, using Brouwers fixed point properties, and Zorns lemma. Also, sufficient conditions for the existence of a unique positive periodic solution has been established using Barbalats lemma and Lyapunovs functional. Numerical responses show that, the phase-flows of the non-autonomous system exhibit an asymptotically stable periodic solution which is globally attractive and trapped in the absorbing region.


Introduction
Mathematical modelling of ecological system has explored robust modifications in terms of the nature of their interactions (i.e., competitive, prey-predator systems, spatio-temporal dynamics, coope-rative systems, patch-diffusion, delay systems and so on), functional responses (i.e., Holling types, Leslie-Gower, Beddington-DeAngelis, and so on) and ecologically perturbative parameters. In prey-predator systems, it is pertinent to assume that all biological and environmental perturbative parameters and state variables are subject to natural fluctuations in time. Thus, the assumption of periodically varying perturbative parameters is a way of making the dynamical system more realistic as compared to constant perturbative parameters. Obviously, periodic variations in the environment and ecologically perturbative parameters are characterized by seasonal effect of weather, food supplies, predation effects, mating durations, time delay due to gestation, and so on.
The qualitative dynamical behaviors of these mathematical models are widely studied in populations of multiple interacting species in the ecosystem. [1] investigated the existence and global attractivity of positive periodic solutions for a Holling II two-prey and one-predator system. Periodic solutions for a three-species Lotka-Volterra food chain model with time delay were studied in [2]. They derived the sufficient conditions for the existence of positive periodic solutions of the system. In the same [3], obtained the necessary and sufficient conditions for existence of periodic solutions of predator-prey dynamical system with Beddington-DeAngelis-type functional response. Existence of periodic solutions for a two-species non-autonomous competitive Lotka-Volterra patch system with time delay was established in [4].
Exploration of these robust dynamical systems requires using topological degree theory, see [5] [6] [7]. In this theory, to prove the existence of solution for a natural abstract formulated IVP, say usually reduces to solving the abstract operator equation, L(X) = N (X) which has some topological degree properties, see [8]. Moreover, results of theorems, and propositions well established via Topological Degree Theory can be numerically simulated using sophisticated dynamical tools (e.g MAPLE) [9] [10].

Model Formulation and Its Invariance Region
The Extended Rosenzwieg-MacArthur Model formulated and studied in [11] is given as: where x1(t), x2(t), and x3(t) are the population densities of the interacting species and r, K, a2, a3, b1, b2, c2, c3, d2 and d3 are positive ecological parameters. In [12] a topologically equivalent dynamical model of system (2.1) was obtained via non-dimensionalization of the state variables as follows: Suppose the ecological parameters are periodic functions, so system (2.2) can be modified to a non-autonomous system as follows: Using the fundamental theorem of calculus, it is easy to see that R 3 + is the invariance region of solutions of system (2.3) satisfying; Thus, the state variables are invariants in the positive octant cone,

Lemma 2 [8]
Let X and Y be two Banach spaces and let L :

Lemma 3 [7]
Let Ω ∈ R n be an open bounded set and L :Ω → R n be a continuous mapping. If p / ∈ L(∂Ω), then the Brouwer degree of L at p relative to Ω is an integer number, denoted by:

Lemma 4 [5]
Let Ω be an open bounded set. Let L be a Fredholm mapping of index zero, and N be L − compact onΩ. Assume ii QN x ̸ = 0, ∀x ∈ DomL ∩ ∂Ω, and

Proposition 1
Assuming that the perturbation parameters of system (2.3) are periodic functions, then system (2.3) has at least one positive periodic solution.
is the phase flows system (2.3), then equipped the spaces, X and Y with the usual Euclidean norm, Define two continuous projectors P : X → X and Q : Y → Y as Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse KP of LP has the form KP : ImL → DomL ∩ KerP , )dτ and Kp(I − Q)N : X → X yields We now seek a forward invariance set K ⊂ X that is convex and compact such that the phase flows Φ(τ ) ⊂ K satisfy the operator equation Lx = tN x, t ∈ (0, 1). Consider and Using Mean-Value Theorem for integral equations, we have that there exists δi where R1, R2, R3 are sufficiently large. Using lemma 1, system (4.4) and proposition (1.6) from [14], the forward invariance region of system (2.3) is as follows: M3] is forward invariance, compact and convex. Using Brouwer fixed point theorem, see [15], the phase flows Φ(τ ) of system (2.3) have at least a fixed point say, (u * , v * , w * ) ∈ X such that Φ(τ ) → (u * , v * , w * ) as τ → ∞. By Zorns lemma, and semi-group properties of phase flows Φ(τ ) of system (2.3), see [16], there exists a maximal element M satisfying then it is easy to claim that Ω is an open bounded set in X, which verifies lemma 4 (i). When u(τ ), v(τ ), w(τ )) T ∈ ∂Ω ∩ KerL = ∂Ω ∩ R 3 ; (u(τ ), v(τ ), w(τ )) T is a constant vector in R 3 with | u | + | v | + | w |= M and the operator equation QN x ̸ = 0 which verifies lemma 4(ii). We now verify lemma 4(iii) using lemma (3) as follows. Define a homotopic mapping, say H(u, v, w; λ) : Moreover, it can be easily shown that the approximated algebraic system (4.5) has a unique fixed point (u * , v * , w * ) ∈ X ⊂ R 3 ifβ >μ. Using homotopy invariance properties of Brouwer's degree, and taking J = I : ImQ → KerL then, Therefore, conditions of lemma (4) are satisfied, and system (2.3) has at least one ω − periodic solution in DomL ∩Ω.

Corollary 1
The is the absorbing region of phase flows of dynamical system (2.3) in Ω.

Proposition 2
Assume the perturbation parameters of dynamical system (2.3) are positive periodic functions, then the dynamical system(2.3) has a unique positive periodic solution, and globally attractive in absorbing region K.
Consider the π−periodic coefficients of system (

Conclusion
This paper has established the necessary conditions for the existence of at least one positive periodic solution of an Extended Rosenzweig-MacArthur tri-trophic food chain model via Brouwers topological degree theory. Also, it has established the sufficient conditions for existence of a unique positive periodic solution of the model using Barbalats lemma and Lyapunovs functional. Consequently, the periodic solution is globally attractive in its invariance region. Thus, this model predicts and depicts a real-life ecological population dynamics as the perturbation parameters assumed periodic oscillations. Its connotes the natural ecological fluctuations.