Global convergence of a reaction–diffusion predator–prey model with stage structure for the predator
Introduction
Stage-structured models have received great attention in recent years (see, for example, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [16], [17], [18], [19], [20], [21], [22]). The pioneering work of Aiello and Freedman [1] on a single species growth model with stage structure represents a mathematically more careful and biologically meaningful formulation approach. In [1], a model of single species population growth incorporating stage structure as a reasonable generalization of the classical logistic model was formulated and discussed. This model assumes an average age to maturity which appears as a constant time delay reflecting a delayed birth of immatures and a reduced survival of immatures to their maturity. The model takes the formwhere ui(t) denotes the immature population density, um(t) represents the mature population density, α > 0 represents the birth rate, γ > 0 is the immature death rate, β > 0 is the mature death and overcrowding rate, τ is the time to maturity. The term αe−γτum(t − τ) represents the immatures who were born at time t − τ and survive at time t(with the immature death rate γ), and therefore represents the transformation of immatures to matures. Aiello and Freedman in model (1.1) assume that the maturation delay τ is know exactly and that all individuals take this amount of time to mature. In [3], Al-Omari and Gourley studied a more general model than model (1.1) by replacing the term αe−γτum(t − τ) with a distribution of maturation times, weighted by a probability density function. They assumed that at time t the number that become mature, per unit time, isThe term αum(t − s) is the number born at time t − s per unit time, and is taken as proportional to the number of mature adults then around. The function f(s) is the probability of taking time s to mature, and e−γs is the probability of an individual born at time t − s still being alive at time t. Individuals becoming mature at time t could have been born at any time prior to this, and the integral totals up the contributions from all previous times. Therefore, the model (1.1) is generalized towhere and f(s) ⩾ 0.
The effect of spatial dispersion on population dynamics has received considerable recent attention. In this situation, the governing equations for the population densities are described by a system of reaction–diffusion equations. An ecological interesting and mathematically challenging problem is to determine whether the time-dependent solution converges to a positive steady state solution, and to which one, if these are multiple, for a given class of initial data (see, for example, [12], [13], [14]).
Motivated by the work on stage-structured competition model by Al-Omari and Gourley [3] and the work on predator–prey model without stage structure by Pao [12], [13], [14], in the present paper, we discuss the following delayed reaction–diffusion Lotka–Volterra type model for prey and adult predator interaction:In problem (1.3), Ω is a bounded domain in with smooth boundary ∂Ω, where ∂/∂ν denotes the outward normal derivative on ∂Ω. The boundary conditions in (1.3) imply that the populations do not move across the boundary ∂Ω. The parameters r1, r2, a11, a12, a22, α and γ are positive constants. u1(t, x) represents the density of the prey population at time t and location x, u2(t, x) denotes the density of the mature predator population at time t and location x, respectively. The data ϕi(t, x) (i = 1, 2) are nonnegative and Hölder continuous and satisfy ∂ϕi/∂ν = 0 in (−∞, 0) × ∂Ω. The model is derived under the following assumptions.
- (A1)
The prey population: the growth of the species is of Lotka–Volterra nature. The parameters r1, a11 and D1 are the intrinsic growth rate, intra-specific competition rate and diffusion rate, respectively.
- (A2)
The predator population: a12, α/a12, r2, and a22 are the capturing rate, conversion rate, death rate and intra-specific competition rate of the mature predator, respectively; γ > 0 is the death rate of the immature predator population, D2 is the diffusion rate of the mature population. The term αu1(t − s, x)u2(t − s, x) is the number born at time t − s and location x per unit time, and is taken as proportional to the number of the prey and mature predator then around. f(s)ds denotes the probability that the maturation time is between s and s + ds with ds infinitesimal, and . e−γs is the probability of an individual born at time t − s still being alive at time t. Individuals becoming mature at time t could have been born at any time prior to this, and the integral totals up the contributions from all previous times.
Following [3], in this paper, for technical reasons we always assume that the kernel f(s) has compact support, that is, f(s) = 0 for all s ⩾ τ, for some τ > 0, and normalized such that . This is also biologically reasonable. In this case, problem (1.3) becomesfor t > 0 and x ∈ Ω, with homogeneous Neumann boundary conditionsand initial conditionswhere ϕi (i = 1, 2) are nonnegative and Hölder continuous with .
In this paper, for system (1.4) we always assume that the following assumption holds:
- (H1)
f(t) is piecewise continuous in [0, τ] and has the property: .
This paper is organized as follows. In the next section, we discuss the existence, uniqueness, positivity and boundedness of solutions of problem (1.4), (1.5), (1.6). By successively modifying the coupled lower–upper solution pairs, sufficient conditions independent of the effect of spatial diffusion are derived for the global convergence of the positive solutions of problem (1.4), (1.5), (1.6). A brief discussion is presented in Section 3.
Section snippets
Global convergence
In this section, we first discuss the existence, uniqueness, positivity and boundedness of problem (1.4), (1.5), (1.6). To do so, we need the following concepts and results. Definition 2.1 A pair of functions in are called coupled upper and lower solutions of system (1.4), (1.5), (1.6) if in and if for all , the following differential inequalities hold:
Discussion
Motivated by the work of Al-Omari and Gourley [3] and Pao [12], [13], [14], in this paper, we incorporated spatial diffusion and stage structure for the predator into a Lotka–Volterra type predator–prey model. By using the coupled upper–lower solution technique, we derived sets of sufficient conditions to respectively guarantee that positive solutions of problem (1.4), (1.5), (1.6) either converge to the unique, positive, uniform equilibrium or to the semi-trivial uniform equilibrium. By
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2012, NeurocomputingCitation Excerpt :In the real-life world, there are many species whose growth rate is not only related to the current number of population, but also to the past one of population [25]. Therefore, in order to simulate more realistic food webs, time delays [25–27] should be considered. Wang et al. [25] proposed a class of three-species Lotka–Volterra mutualism models with diffusion and delay effects.
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The first author’s work was supported by the National Natural Science Foundation of China (No. 10471066).