Traveling waves for a generalized Holling-Tanner predator-prey model
We study traveling wave solutions for Holling-Tanner type predator-prey models, where the predator equation has a singularity at zero prey population. The traveling wave solutions here connect the prey only equilibrium (1, 0)with the unique constant coexistence equilibrium (u∗, v∗). First, we give a sharp existence result on weak traveling wave solutions for a rather general class of predator-prey systems, with minimal speed explicitly determined. Such a weak traveling wave (u(ξ), v(ξ))connects (1, 0)at ξ=-∞but needs not connect (u∗, v∗)at ξ=∞. Next we modify the Holling-Tanner model to remove its singularity and apply the general result to obtain a weak traveling wave solution for the modified model, and show that the prey component in this weak traveling wave solution has a positive lower bound, and thus is a weak traveling wave solution of the original model. These results for weak traveling wave solutions hold under rather general conditions. Then we use two methods, a squeeze method and a Lyapunov function method, to prove that, under additional conditions, the weak traveling wave solutions are actually traveling wave solutions, namely they converge to the coexistence equilibrium as ξ→∞.
The Fisher-KPP equation over simple graphs: varied persistence states in river networks
In this article, we study the dynamical behaviour of a new species spreading from a location in a river network where two or three branches meet, based on the widely used Fisher-KPP advection-diffusion equation. This local river system is represented by some simple graphs with every edge a half infinite line, meeting at a single vertex. We obtain a rather complete description of the long-time dynamical behaviour for every case under consideration, which can be classified into three different types (called a trichotomy), according to the water flow speeds in the river branches, which depend crucially on the topological structure of the graph representing the local river system and on the cross section areas of the branches. The trichotomy includes two different kinds of persistence states, and the state called "persistence below carrying capacity" here appears new.
The parabolic logistic equation with blow-up initial and boundary values
In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values... We study the existence and uniqueness of positive solutions, and their asymptotic behavior near the parabolic boundary. Under the extra condition that b(x, t) ≥ c(T - t)⁰d(x,∂Ω)ᵝ on Ω x [0,T) for some constants c > 0,Ɵ > and β > -2, we show that such a solution stays bounded in any compact subset of Ω as t increases to T, and hence solves the equation up to t = T.
Period-parabolic Logistic Equation with Spatial and Temporal Degeneracies
A basic goal of theoretical ecology is to understand how the interactions of individual organizations with each other under a certain inhabiting environment determine the spatiotemporal structure of distribution of populations. The diffusive logistic equation, which describes the spatial and temporal distribution of the population density of a single species, is one of the fundamental reaction-diffusion equation models in population biology. In this thesis, we are concerned with the periodic logistic equation with homogeneous Neumann boundary conditions. The theory for this basic case forms the foundation for further investigation of multispecies problems. The thesis consists of four chapters.
Sharp spatiotemporal patterns in the diffusive time-periodic logistic equation
To reveal the complex influence of heterogeneous environment on population systems, we examine the asymptotic profile (as ϵ → 0) of the positive solution to the perturbed periodic logistic equation... We show that the temporal degeneracy of 'b' induces sharp spatiotemporal patterns of the solution only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in 'b', the spatial degeneracy always induces sharp spatiotemporal patterns, though they differ significantly according to whether temporal degeneracy is also present.
Effect of a protection zone in the diffusive Leslie predator-prey model
In this paper, we consider the diffusive Leslie predator-prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone Ω₀ for the prey is introduced. As in earlier work [Y. Du, J. Shi. A diffusive predator-prey model with a protection zone. J. Differential Equations 229 [2006] 63-91: Y. Du. X. Liang. A diffusive competition model with a protection zone. J. Differential Equations 244 (2008) 61-86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over Ω₀ and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when Ω₀ is above the critical patch size. However, our research here reveals significant difference of the model's behavior from the predator-prey model studies in [Y. Du, J. Shi, A diffusive predator-prey model with a protection some, J. Differential Equations 229 (2006) 63-91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations.
Qualitative Analysis of a Cooperative Reaction-Diffusion System in a Spatiotemporally Degenerate Environment
In this paper, we are concerned with the cooperative system in which ∂tu − Δu = μu + α(x, t)v − a(x, t)up and ∂tv − Δv = μv + β(x, t)u − b(x, t)vq in Ω × (0,∞); (∂ν u, ∂νv) = (0, 0) on ∂Ω×(0,∞); and (u(x, 0), v(x, 0)) = (u0(x), v0(x)) > (0, 0) in Ω, where p, q > 1, Ω ⊂ RN (N ≥ 2) is a bounded smooth domain, α, β > 0 and a, b ≥ 0 are smooth functions that are T-periodic in t, and μ is a varying parameter. The unknown functions u(x, t) and v(x, t) represent the densities of two cooperative species. We study the long-time behavior of (u, v) in the case that a and b vanish on some subdomains of Ω × [0, T]. Our results show that, compared to the nondegenerate case where a, b > 0 on Ω × [0, T], such a spatiotemporal degeneracy can induce a fundamental change to the dynamics of the cooperative system.
The periodic logistic equation with spatial and temporal degeneracies
In this article, we study the degenerate periodic logistic equation with homogeneous Neumann boundary conditions... Our analysis leads to a new eigenvalue problem for periodic-parabolic operators over a varying cylinder and certain parabolic boundary blow-up problems not known before. The investigation in this paper shows that the temporal degeneracy causes a fundamental change of the dynamical behavior of the equation only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in the equation, the spatial degeneracy always induces fundamental changes of the behavior of the equation, though such changes differ significantly according to whether or not there is temporal degeneracy.
Traveling wave solutions of a class of multi-species non-cooperative reaction–diffusion systems
In this paper we establish a sharp existence result on weak traveling wave solutions for a general class of multi-species reaction–diffusion systems. Moreover, the minimal speed of the traveling waves is explicitly determined. Such a weak traveling wave solution connects the predator-free equilibrium point E0 at x = −∞ but needs not to connect the coexistence equilibrium E∗ at x = ∞. We apply this result to three important non-cooperative systems: the classical diffusive SIS system for the spread of infectious disease, a predator–prey system with age structure and a generalised Lotka–Volterra predator–prey system of one predator species feeding on n prey species, and prove with the aid of Lyapunov functions and the LaSalle invariance principle that their weak traveling wave solutions are actually traveling wave solutions that connect E ∗ at x = ∞. For the SIS system and the generalised Lotka–Volterra predator–prey system, we develop additional techniques to establish the boundedness of their weak traveling wave solutions before applying the LaSalle's invariance principle.
A diffusive logistic model with a free boundary in time-periodic environment
We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in Du and Lin (2010), Du and Guo (2011). In both cases, a spreading-vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading-vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods.