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1.
Some uniqueness and exact multiplicity results for a predator-prey model   总被引:6,自引:0,他引:6  
In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large reveal interesting contrast with that for the well-studied case , i.e., the classical Lotka-Volterra predator-prey model.

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2.
Equations with non-local dispersal have been widely used as models in biology. In this paper we focus on logistic models with non-local dispersal, for both single and two competing species. We show the global convergence of the unique positive steady state for the single equation and derive various properties of the positive steady state associated with the dispersal rate. We investigate the effects of dispersal rates and inter-specific competition coefficients in a shadow system for a two-species competition model and completely determine the global dynamics of the system. Our results illustrate that the effect of dispersal in spatially heterogeneous environments can be quite different from that in homogeneous environments.  相似文献
3.
We consider reaction-diffusion-advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions.  相似文献
4.
We consider the Shigesada-Kawasaki-Teramoto cross-diffusion model for two competing species. If both species have the same random diffusion coefficients and the space dimension is less than or equal to three, we establish the global existence and uniform boundedness of smooth solutions to the model in convex domains. This extends some previous works of Kim [12 Kim, J.U. (1984). Smooth solutions to a quasilinear system of diffusion equations for a certain population model. Nonlinear Anal. 8:11211144.[Crossref], [Web of Science ®] [Google Scholar]] and Shim [21 Shim, S.-A. (2002). Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Eqs. 185:281305.[Crossref], [Web of Science ®] [Google Scholar]] in one dimensional space.  相似文献
5.
We first investigate in a logistic model the effects of migration and spatial heterogeneity of the environment on the total population size at equilibrium of a single species. Our study shows that (i) the total population size is maximized at some intermediate migration rate, and hence is a non-monotone function of the migration rate; (ii) heterogeneity of the environment increases the population size. In the second part of this paper, these findings are applied to ecological invasions. For a two-species Lotka-Volterra competition model with migration, we show that (i) without migration, the invading species eliminates the resident species at every point of the habitat, whereas when migration is present, for certain ranges of migration rates the invader may be eliminated when it is rare; and (ii) without migration, the two species can coexist at every point of the habitat, whereas when migration is present, for some ranges of migration rates one of the species is extinguished for all initial conditions.  相似文献
6.
The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus without dominance is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape (i.e., a bounded, open domain in Rd). The selection coefficients depend on position; the drift and diffusion coefficients may depend on position. The primary focus of this paper is the dependence of the evolution of the gene frequencies on λ, the strength of selection relative to that of migration. It is proved that if migration is sufficiently strong (i.e., λ is sufficiently small) and the migration operator is in divergence form, then the allele with the greatest spatially averaged selection coefficient is ultimately fixed. The stability of each vertex (i.e., an equilibrium with exactly one allele present) is completely specified. The stability of each edge equilibrium (i.e., one with exactly two alleles present) is fully described when either (i) migration is sufficiently weak (i.e., λ is sufficiently large) or (ii) the equilibrium has just appeared as λ increases. The existence of unexpected, complex phenomena is established: even if there are only three alleles and migration is homogeneous and isotropic (corresponding to the Laplacian), (i) as λ increases, arbitrarily many changes of stability of the edge equilibria and corresponding appearance of an internal equilibrium can occur and (ii) the conditions for protection or loss of an allele can both depend nonmonotonically on λ. Neither of these phenomena can occur in the diallelic case.  相似文献
7.
A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.  相似文献
8.
The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape (i.e., a bounded, open domain in Rd). The selection coefficients depend on position and may depend on the gene frequencies; the drift and diffusion coefficients may depend on position. Sufficient conditions are given for the global loss of an allele and for its protection from loss. A sufficient condition for the existence of at least one internal equilibrium is also offered, and the profile of any internal equilibrium in the zero-migration limit is obtained.  相似文献
9.
We study the effects of advection along environmental gradients on logistic reaction-diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. It is also assumed that the boundary acts as a reflecting barrier to the population. We show that the effects of such advection depend crucially on the shape of the habitat of the population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is always beneficial to the population, while such advection could be harmful for certain non-convex habitats.  相似文献
10.
In this paper, the formats of Julia sets for a class of nonlinear complex dynamic systems with variable coefficients were studied under certain conditions. For the complex dynamic systems in piecewise cases, we proposed some methods to control the forms of their Julia sets and stable domains analytically. What’s more, we illustrated that our methods worked well by computational simulations. Our work provides a better understanding about how to control the Julia sets of certain complex dynamic systems.  相似文献
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