Permanence And Periodicity Of Two Type Predator-prey Model With Holling-Ⅲ Functional Response

Traditional Lotka-Volterra type predator-prey with Holling-â…¢functionalresponse has received great attention from both theoretical and mathematical biologists, and has been well studied. The standard Lotka-Volterra typemodel is built by assuming that the per capita rate of predation depends onthe prey numbers only. Recently, the traditional prey-dependent predatorprey model has been challenged by several biologists(see, for example [4-8]).There is growing explicit biological and physiological evidence [4-8] that insome situations, especially when predator have to search for food (and therefore have to share or compete for food), a more suitable general predator-preymodel should be based on the ratio-dependent theory. This roughly statesthat the per capita predator growth rate should be a function of the ratioof the prey to predator abundance. This is strongly supported by numerousfield and laboratory experiments and observations[4, 6, 8]. In Chapterâ… ofthis paper, a periodic and delayed ratio-dependent predator-prey system withHolling typeâ…¢functional response and stage structure for both prey andpredator is investigated, sufficient conditions are derived for the permanenceand existence of positive periodic solution of system (1.1.3). In Chapterâ…¡,a ratio-dependent predator-prey system with Holling-â…¢functional responseand impulsive effect is considered. Conditions for permanence is establishedvia the method of comparison involving multiple Lyapunov functions. Theexistence of a positive periodic solution is also studied by the bifurcationtheory.Chapterâ… Permanence and periodicity of a delayedratio-dependent predator-prey model with Holling typefunctional response and stage structure In the chapter, we consider the following delayed ratio-dependent predatorprey system with Holling typeâ…¢functional response and stage structure.where x₁(t) and x₂(t) denote the densities of immature and mature individual preys at time t, respectively; y₁(t) and y₂(t) represent the densities of immature and mature individual predators at time t, respectively.α₁(t),α₂(t),β₁(t),β₂(t),γ₁(t),γ₂(t), andα₁(t) are continuously positive periodic functions with periodω, and x₂²(t)/(m²y₂²(t)+x₂²(t)) here denotes themature predator response function, which reflects the capture ability of themature predator. The model is derived under the following assumptions(seeChapterâ… (H₁)-(H₂)).The initial conditions for system (1.1.3) take the form ofwhereÏ„=max{Ï„₁,Ï„₂}, (φ₁(θ),φ₂(θ),ψ₁(θ),ψ₂(θ))∈C([-Ï„,0], R₊₀⁴), the Banach space of continuous functions mapping the interval [-Ï„, 0] into R₊₀⁴,where we define R₊₀⁴={(x₁, x₂, x₃, x₄): x_i≥0, i=1, 2, 3, 4},and the interior of R₊⁴, R₊⁴={(x₁, x₂, x₃, x₄): x_i＞0, i=1, 2, 3, 4}. For continuity of initial conditions, we requireWe adopt the following notations throughout Chapterâ… : (?)=1/ωintegral from n=0 toωf(t)dt, f^L=(?) |f(t)|, f^M=(?) |f(t)|,where f is a continuousω-periodic function.We get main conclusions about system (1.1.3) in the followingTheorem 1.2.1. System (1.1.3) with initial conditions (1.1.4) and(1.1.5) is permanent provided that(H₃) 2mα₁^Le^{-γ₁MÏ„₁}＞α₁^M,β₂^M＜α₂^Le^{-γ₂MÏ„₂}＜2β₂^M.Theorem 1.2.2. Adult predator population will go to extinction ifTheorem 1.3.1. Let (H₃) hold. Then system (1.1.3) with initial conditions (1.1.4) and (1.1.5) has at least one strictly positiveω-periodic solution.Chapterâ…¡Persistence and periodicity of a predator-preysystem with Holling-â…¢functional response and impulsiveeffectIn the chapter, we consider system (see(2.1.2))Whereâ–³x_i(t)=x_i(t⁺)-x_i(t), i=1, 2, 0≤p₁＞1(0≤p₂＜1) representsthe fraction of pest (predator) which dies due to the pesticide, q≥0 is the release amount of predator at t=nT, n∈Z₊ and Z₊={1, 2,…}, T isthe period of the impulsive effect, and a, b, c,α,β,κare positive constants.That is, we can use a combination of biological (periodic releasing naturalenemies) and chemical (spraying pesticide) tactics that eradicates the pestto extinction, and show the efficiency of IPM strategy.We get main conclusions about system (2.1.2) in the followingTheorem 2.3.1. Let (x(t),y(t)) be any solution of (2.1.2). Then(0, y^*(t)) is globally asymptotically stable if T＜(-ln(1-p₁))/αTheorem 2.3.2. There exists a constant M＞0 such that x(t)≤M, y(t)≤M for each solution (x(t), y(t)) of system (2.1.2) with all t largeenough.Theorem 2.3.3. System (2.1.2) is permanent provided T＞(-ln(1-p₁))/αholds true.Theorem 2.4.1. System (2.1.2) has a positive periodic solution if T＞T₀=-(ln(1-p₁))/αand is closing to T₀.