Yongyan Huang, Fuyi Li, Junping Shi
We study the reaction-diffusion Lotka-Volterra predator-prey model with Dirichlet boundary condition. In the case of equal diffusion rates and equal growth rates, the synchronized steady state solution is proved to be locally asymptotically stable.
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This paper is concerned with a diffusive predator-prey model with predator-taxis and prey-taxis. Based on the Schauder fixed point theorem, we prove the global existence, uniqueness and boundedness of the classical solutions under the conditions that the predator-taxis and prey-taxis effects are weak enough.
The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary is considered. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behavior of the global solution. Some sufficient conditions for both spreading and vanishing are established.
This paper investigates the large time behaviour of a three species reaction-diffusion system, modelling the spatial invasion of two predators feeding on a single prey species. In addition to the competition for food, the two predators exhibit competitive interactions and under some parameter conditions ($μ>0$), they can also be considered as two mutants. When mutations occur in the predator populations, the spatial spread of invasion takes place at a definite speed, identical for both mutants. When the two predators are not coupled through mutation, the spreading behaviour exhibits a more com…
An age-structured predator-prey system with diffusion and Holling-Tanner-type nonlinearities is considered. Regarding the intensity of the fertility of the predator as bifurcation parameter, we prove that a branch of positive coexistence steady states bifurcates from the marginal steady state with no prey. A similar result is obtained when the fertility of the prey varies.
We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for preys. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of predators can be directed towards region with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the …
We investigate the traveling wave solutions of a three-species system involving a single predator and a pair of strong-weak competing preys. Our results show how the predation may affect this dynamics. More precisely, we describe several situations where the environment is initially inhabited by the predator and by either one of the two preys. When the weak competing prey is an aboriginal species, we show that there exist traveling waves where the strong prey invades the environment and either replaces its weak counterpart, or more surprisingly the three species eventually co-exist. Furthermor…
In this paper, we study the Lotka-Volterra prey-predator models consisting of two species on finite connected graphs under Neumann condition and the condition that there is no boundary condition. We establish the global stability of the unique constant equilibrium solution of each parabolic system.
The time-global unique solvability on the reaction diffusion equations for prey-predator models with density-dependent inhibitor and dormancy on predators is established. The crucial step of the proof is to construct time-local non-negative classical solutions. To do so, new successive approximation and theories of time-evolution operators are used. Due to the maximum principle, the solutions are extended time-globally. Via analysis on the corresponding ordinary differential equations, invariant regions and asymptotic behaviors of solutions are also investigated.
We study a system of elliptic equations with strong competition and an arbitrary large number of components. The system is related to a model of predators and prey, with a single and where several predators compete with each other. In this paper we derive regularity estimates of the solutions that are independent of the number of components (i.e., groups of predators) and the strength of competition between the components.
In this paper we study a ratio-dependent predator-prey model with a free boundary causing by both prey and predator over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy, namely, as t goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and die out eventually. Then the criteria governing spreading and vanishing are obtained. Finally, when spreading occurs, we provide some estimates to the asymptotic spreading speed of h(t).
In this paper we study a free boundary problem for a ratio-dependent predator-prey system in one space dimension, with the free boundary only caused by the prey. The long time behaviors of solution are discussed. Then we establish a spreading-vanishing dichotomy and the criteria for spreading and vanishing. Finally, when spreading occurs, an accurate limit of the asymptotic speed of h(t) is provided.
We study a spatial (two-dimensional) Rosenzweig-MacArthur model under the following assumptions: $(1)$ prey movement follows a nonlinear diffusion, $(2)$ preys have a refuge zone (sometimes called "protection zone") where predators cannot enter, (3) predators move following linear diffusion. We present a bifurcation analysis for the system that shows the existence of positive solutions at the steady state. We complement the theoretical results with numerical computations and compare our results with those obtained in the case of having linear diffusion for the prey movement. Our results show t…
Traveling Wave Solutions For A Singular Diffusive Prey-Predator Model With Nonlocal Dispersal
本文研究了具有非局部扩散的奇异捕食者-猎物模型，证明了存在连接捕食者无状态和共存状态的正行波解，并确定了波速的范围。
For a stationary system representing prey and $N$ groups of competing predators, we show classification results about the set of positive solutions. In particular, we show that if the number of components $N$ is too large or if the competition between different groups is too small, then the system has only constant solutions, which we then completely characterize.
The global stability of the nonhomogeneous positive steady state solution to a diffusive Holling-Tanner predator-prey model in a heterogeneous environment is proved by using a newly constructed Lyapunov function and estimates of nonconstant steady state solutions. The techniques developed here can be adapted for other spatially heterogeneous consumer-resource models.
This paper focus on the diffusive eco-epidemiological prey-predator model with infectious diseases in prey, and with the homogeneous Neumann and Dirichlet boundary conditions, respectively. When boundary conditions are homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative constant equilibrium solutions. The results show that such a problem has neither periodic solutions nor Turing patterns. When boundary conditions are homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of …
To understand the spreading and interaction of prey and predator, in this paper we study the dynamics of the diffusive Lotka-Volterra type prey-predator model with different free boundaries. These two free boundaries, which may intersect each other as time evolves, are used to describe the spreading of prey and predator. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behaviors of global solution. Some sufficient conditions for spreading and vanishing are established. When spreading occurs, we provide the more accurate limits of (u,v) as t\to\infty,…
In this paper, we investigate so-called forced wave solutions of a three components reaction-diffusion system from population dynamics. Our system involves three species that are respectively two competing preys and one predator; moreover, the competition between both preys is strong, i.e. in the absence of the predator, one prey is driven to extinction and the other survives. Furthermore, our problem includes a spatio-temporal heterogeneity in a moving variable that typically stands as a model for climate shift. In this context, forced waves are special stationary solutions which are expected…
In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator-prey systems with density-dependent preytaxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusion) and mobility (preytaxis) of the predator are correlated through a prey density dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on paramet…
Global dynamics of a predator-prey model with alarm-taxis
本文针对警报性趋化捕食者-被捕食者模型，提出利用略高阻尼幂的新梯度估计方法，证明二维Neumann边界下经典解有界，并通过Lyapunov泛函证明共存稳态指数渐近稳定。
We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider both the cases of nonlocal and local dispersal. In both these situations, we first prove the existence of forced waves, which are positive stationary solutions in the moving frames of the climate change, of either front or pulse type. Then we address the persistence or extinction of the prey and the predator separately in various moving frames, and achieve a co…
We consider a three-species predator-prey system involving two competing predators and one prey. The species diffuse with nonlocal dispersal kernels with possibly non-compact support, and they interact in a heterogeneous environment moving with a positive forced speed such that the environment is favorable to the prey in the absence of predators far ahead of the shifting boundary and it is unfavorable far behind. Such systems arise in the modeling of population dynamics under the effect of a shifting environment, such as climate change. We show on the one hand the existence of waves connecting…
Asymptotic spreading speeds of a road-field reaction-diffusion predator-prey model
本文研究道路-田野反应扩散捕食者-被捕食者模型的渐近传播速度，通过逐点比较与截断系统约化，刻画了捕食者与被捕食者的入侵速度及其方向依赖性。
In any reaction-diffusion system of predator-prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To save the endangered prey species, some human interference is useful, such as creating a protection zone where the prey could cross the boundary freely but the predator is prohibited from entering. This …
This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension $N=1,2$ and for the parabolic- -parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that pre…
Spreading speeds for prey--predator systems in a shifting environment: a short proof
研究移动环境中捕食-被捕食反应扩散系统中捕食者组分的传播速度，通过点态估计将系统解与Fisher-KPP方程比较，得到显式传播速度，并发现非单调移动环境中的锁定和非局部拉动现象。
In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the …
We study a mathematical model of environments populated by both preys and predators, with the possibility for predators to actively compete for the territory. For this model we study existence and uniqueness of solutions, and their asymptotic properties in time, showing that the solutions have different behavior depending on the choice of the parameters. We also construct heterogeneous stationary solutions and study the limits of strong competition and abundant resources. We then use these information to study some properties such as the existence of solutions that maximize the total populatio…
On the effects of protection zone and directed population flux in prey-predator dynamics
本文研究了保护区和定向种群流对捕食者-猎物动态的影响，揭示了强定向移动如何诱导共存稳态的多样性。
The Influence of Exclusion Zones on the Coexistence of Predator and Prey with an Allee Effect
本文研究了排斥区对具有阿勒效应的捕食者-猎物共存的影响，通过反应扩散模型展示排斥区如何缓解过度捕食导致的物种灭绝，并探讨了捕食者领地大小对种群动态的影响。
In this paper, we investigate the effect of dispersal and advection on the dynamics of a predator-prey model. More precisely, we show that the linear stability of the semi-trivial steady state is determined by the dispersal rate, the mortality rate of the predator and the advection rate. We point out that compared to homogeneous intrinsic growth rate and carrying capacity, the case in this paper is more complicated. This work gives a investigation to an open problem proposed by Nie et al. in \cite{NWW} by considering a more general model, and then, can be seen as a further development of their…
Fast reaction limit for a Leslie-Gower model including preys, meso-predators and top-predators
本研究针对含猎物、两类捕食者的Leslie-Gower反应扩散模型，提出五方程简化模型，证明其快速反应极限可还原原模型零阶项且会出现交叉扩散，还分析了新系统平衡点并证明了一维二维下的部分收敛性。
The role of predator evasion mediated by chemical signaling is studied in a diffusive prey-predator model when prey-taxis is taken into account (model A) or not (model B) with taxis strength coefficients $χ$ and $ξ$ respectively. In the kinetic part of the models it is assumed that the rate of prey consumption includes functional responses of Holling, Bedington-DeAngelis or Crowley-Martin. Existence of global-in-time classical solutions to model A is proved in space dimension n=1 while to model B for any $n\geq 1$. The Crowley-Martin response combined with bounded rate of signal production pre…
Mathematical analysis of a stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises
本文对一个带有趋食性和噪声的捕食者-猎物非线性随机反应扩散系统进行数学分析，通过 Faedo-Galerkin 逼近等方法证明了弱鞅解的存在性与非负性，并利用对偶技术证明了解的唯一性。
In this paper, we study the dynamics of the ratio-dependent type prey-predator model with different free boundaries. The two free boundaries, determined by prey and predator respectively, implying that they may intersect each other as time evolves, are used to describe the spreading of prey and predator. We mainly investigate the longtime behaviors of predator and prey. Then some sufficient conditions for spreading and vanishing are given. In addition, when spreading occurs, some estimates of asymptotic spreading speeds of u, v and asymptotic speeds of h, g are provided.
In this paper we investigate some free boundary problems for the Lotka-Volterra type prey-predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the entire space as time $t$ goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when …
Exploring unique dynamics in a predator-prey model with generalist predator and group defence in prey
本研究提出一个具有泛化捕食者和猎物群体防御的捕食者-猎物模型，发现有限时间爆破与淬灭现象，并证明两者时间相同，同时通过数值分析揭示丰富的分岔和极限环，最后提出延迟版本模型以保证全局解存在。
In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of prey by the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published results show that population structuring and rapid evolution in such predator-prey…
High multiplicity and global structure of coexistence states in a predator-prey model with saturation
本文研究具有饱和项的扩散捕食-被捕食模型，证明在适当参数范围内，足够大的饱和率可导致任意多个共存态，并刻画了饱和率趋于无穷时极限系统中共存态的全局结构。
Persistence of solutions in a nonlocal predator-prey system with a shifting habitat
本文针对非局部扩散捕食-被捕食系统在移动环境中的传播性质，利用先验估计、Arzela-Ascoli定理和对角线提取过程，解决了Choi等人遗留的中间移动坐标系中持续生存问题，从而在非局部扩散情形下得到完整图像。
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