On the Asymptotic Behavior of Solutions of a Reacting–Diffusing System: A two Predators–One Prey Model

Hsiao, Ling; Xin, Zhen

doi:10.1137/0518050

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…It has been generalized to Volterra integral equations by Miller and Sell (1970), e.g. Theorems 1.2 and 1.3 are often applied to show that the solutions of population dynamic (notably chemostat) models converge to an equilibrium (Theorem 1.2: Hsu 1981, Hsiao et al 1987, Cushing 1989Theorem 1.3: Busenberg and Iannelli 1985). Theorem 1.3 has also triggered some research on almost periodic solutions of asymptotically autonomous ODEs in the plane (Utz and Waltman 1967;Grimmer 1968).…”

Section: Introductionmentioning

confidence: 99%

Convergence results and a Poincar�-Bendixson trichotomy for asymptotically autonomous differential equations

1992

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Section: Introductionmentioning

confidence: 99%

Convergence results and a Poincar�-Bendixson trichotomy for asymptotically autonomous differential equations

1992

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“…model of [9] was studied by us in [13]. As for the studies on three species reactiondiffusion systems of predator-prey model, the readers can see [4,5]. 409/124,1bl7 2.…”

Section: In~~ducti~nmentioning

confidence: 99%

A reaction-diffusion system of a competitor-competitor-mutualist model

Zheng

1987

Journal of Mathematical Analysis and Applications

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We investigate the homogeneous Dirichlet problem and Neumann problem to a reactiondiffusion system of a competitor-competitor-mutualist model. The existence, uniqueness, and boundedness of the solutions are established by means of the comparison principle and the monotonicity method. For the Dirichlet problem, we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities. For the Neumann problem, we analyze the contant equilibrium solutions and their stabilities. The main method used in studying of the stabilities is the spectral analysis to the linearized operators. The O.D.E. problem to the same model was proposed and studied by B.

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“…The dynamics of two‐species predator–prey systems has been widely studied (see, e.g., previous studies 1–10 ). However, the investigation of one prey–two predators systems seems to exhibit much richer characteristics (see previous studies 11‐19 ).…”

Section: Introductionmentioning

confidence: 99%

“…If the predators and the prey are confined to a bounded domain Ω with smooth boundary and their densities are spatially inhomogeneous, diffusion terms are added to the reaction terms. Hsiao et al 13 formulated a one prey–two predators model with Holling‐type II functional response in the form

({, \begin{matrix} array \frac{\partial u}{\partial t} = d_{1} Δ u + u (a_{0} - a_{1} u) - \frac{c_{1} u v}{u + m_{1}} - \frac{c_{2} u w}{u + m_{2}} \\ array \\ array \frac{\partial v}{\partial t} = d_{2} Δ v + v (- b_{1} + k_{1} \frac{c_{1} u}{u + m_{1}}) \\ array \\ array \frac{\partial w}{\partial t} = d_{3} Δ w + w (- b_{2} + k_{2} \frac{c_{2} u}{u + m_{2}}), \\ array \end{matrix})

where u , v , and w denote the population densities of the prey and the i th predator (

i = 1,2 false)

, respectively,

\frac{c_{i} u}{u + m_{i}}

i = 1,2

, is the functional response, d i ,

i = 1,2, 3

, is the diffusion coefficient.…”

Section: Introductionmentioning

confidence: 99%

A diffusive two predators–one prey model on periodically evolving domains

Montano

Lisena

2021

Math Methods in App Sciences

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The paper deals with a diffusive two predators–one prey model with Holling‐type II functional response. We assume that the density of prey and predators is spatially inhomogeneous on a periodically evolving domain and is subject to homogeneous Neumann boundary conditions. We focus on the case in which all populations have periodic logistic growth, if isolated, and no competition occurs between predators. Our main purpose is to study the asymptotic properties of the solutions of this reaction–diffusion model. More specifically, suitable conditions, depending on the domain evolution function and the space dimension, are introduced leading to the extinction of one predator and the stable coexistence of the surviving predator and its prey. Their density, as time tends to infinity, tends to the periodic solution of the corresponding kinetic predator–prey model. Finally, the autonomous model on a fixed domain is treated.

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On the Asymptotic Behavior of Solutions of a Reacting–Diffusing System: A two Predators–One Prey Model

Cited by 5 publications

References 11 publications

Convergence results and a Poincar�-Bendixson trichotomy for asymptotically autonomous differential equations

Convergence results and a Poincar�-Bendixson trichotomy for asymptotically autonomous differential equations

A reaction-diffusion system of a competitor-competitor-mutualist model

A diffusive two predators–one prey model on periodically evolving domains

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