Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case

We prove the non-existence of non-constant positive steady state solutions of two reaction-diffusion predator-prey models with Holling type-II functional response when the interaction between the predator and the prey is strong. The result implies that the global bifurcating branches of steady state solutions are bounded loops.     

Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I

To capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, Allen et al. in [L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A 21 (1) (2008) 1-20] proposed a spatial SIS (susceptible-infected-susceptible) reaction-diffusion model, and studied the existence, uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so-called low-risk subhabitat is created. In this work, we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease, which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases. Consequently, our results reveal that, in order to eliminate the infected population at least in low-risk area, it is necessary that one will have to create a low-risk subhabitat and reduce at least one of the diffusion rates to zero. In this case, our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population; moreover, once the spatial environment is modified to include a low-risk subhabitat, the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones.     

Positive solutions for a diffusive Bazykin model in spatially heterogeneous environment

In this paper, we investigate a diffusive Bazykin model in a spatially heterogeneous environment. We obtain some results on nonexistence and existence of positive solutions of the model. Moreover, the asymptotic behavior of positive solutions with respect to certain parameters is also studied.     

A diffusive competition model with a protection zone

This paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor. Our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor, the weak species almost always survives, but it cannot survive when the protection zone is below the critical size and its competitor is strong enough. While this is the main feature of the model, the actual dynamical behavior of the reaction-diffusion system is more complicated. The key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady-state solutions. Our mathematical analysis shows that, compared with the predator-prey model discussed in [Yihong Du, Junping Shi, A diffusive predator-prey model with a protect zone, J. Differential Equations 226 (2006) 63-91], the protection zone has some essentially different effects on the fine dynamics of the competition model.     

Realization of prescribed patterns in the competition model

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D₁,…,D_m of a given spatial domain Ω in R^N, if d₁,d₂,a₁,a₂,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D?D₁∪?∪D_m and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model

     

Strategy and stationary pattern in a three-species predator-prey model

In this paper, we study a strongly coupled system of partial differential equations which models the dynamics of a two-predator-one-prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey's strategy. We demonstrate the emergence of stationary patterns for this system, and show that it is due to the cross diffusion that arises naturally in the model. As far as the authors are aware, this is the first example of stationary patterns in a predator-prey system arising solely from the effect of cross diffusion.     

Turing pattern of the Oregonator model

The Oregonator model is the mathematical dynamics which describes the Field-Körös-Noyes mechanics of the famous Belousov-Zhabotinskii? reaction. In this work, we establish some fundamental analytic properties of this dynamics and its corresponding steady state. Under various conditions on the parameters and the size of the reactor, we examine the existence and non-existence of non-constant steady states. In particular, for some properly chosen parameter ranges, we prove the occurrence of the Turing pattern generated by this Oregonator model. Our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the Turing pattern. Our mathematical analysis mainly relies on a priori estimates and the topological degree argument.     

A diffusive predator-prey model in heterogeneous environment

In this paper, we demonstrate some special behavior of steady-state solutions to a predator-prey model due to the introduction of spatial heterogeneity. We show that positive steady-state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of the behavior of our model from that of the classical Lotka-Volterra model that seem to arise only in the heterogeneous case.     

On positive solutions concentrating on spheres for the Gierer-Meinhardt system

We consider the stationary Gierer-Meinhardt system in a ball of R^N:

     

Loops and branches of coexistence states in a Lotka-Volterra competition model

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.     

A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.     

A strongly coupled predator–prey system with non-monotonic functional response

The predator–prey system with non-monotonic functional response is an interesting field of theoretical study. In this paper we consider a strongly coupled partial differential equation model with a non-monotonic functional response—a Holling type-IV function in a bounded domain with no flux boundary condition. We prove a number of existence and non-existence results concerning non-constant steady states (patterns) of the underlying system. In particular, we demonstrate that cross-diffusion can create patterns when the corresponding model without cross-diffusion fails.     

Advection-mediated competition in general environments

We consider a reaction–diffusion–advection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some non-degeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is one-dimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this non-standard system is obtained when one of the diffusion rates is large.     

The steady states of a non-cooperative model of nuclear reactors

This paper characterizes the existence of coexistence states in a reaction-diffusion model arising in the theory of nuclear reactors. From a mathematical point of view, the importance of this model relies upon the fact that the associated variational systems are of non-cooperative type and, consequently, the comparison techniques available for cooperative systems fail to work out. Although in higher spatial dimensions the dynamics of the model might be rather involved, by the absence of limitations for the number of steady states, we can prove the uniqueness of the steady state in the one-dimensional prototype model. Our results complement and eventually sharpen the findings of Arioli [G. Arioli, Long term dynamics of a reaction-diffusion system, J. Differential Equations 235 (2007) 298-307].     

Turing patterns of a strongly coupled predator-prey system with diffusion effects

The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator-prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction-diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction-diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability.     

Clustered spots in the FitzHugh-Nagumo system

We construct clustered spots for the following FitzHugh-Nagumo system:

     

Bound states of the Schrödinger-Newton model in low dimensions

We prove the existence of quasi-stationary symmetric solutions with exactly n≥0 zeros and uniqueness for n=0 for the Schrödinger-Newton model in one dimension and in two dimensions along with an angular momentum m≥0. Our result is based on an analysis of the corresponding system of second-order differential equations.     

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge

We study a predator-prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant m∈(0,1], which provides a condition for protecting (1−m)u of prey u from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.