Interspecific density-dependent model of predator–prey relationship in the chemostat

The objective of this study is to analyze a model of competition for one resource in the chemostat with general interspecific density-dependent growth rates, taking into account the predator–prey relationship. This relationship is characterized by the fact that the prey species promotes the growth of the predator species which in turn inhibits the growth of the first species. The model is a three-dimensional system of ordinary differential equations. With the same dilution rates, the model can be reduced to a planar system where the two models have the same local and even global behavior. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. Using the nullcline method, we present a geometric characterization of the existence and stability of all equilibria showing the multiplicity of coexistence steady states. The bifurcation diagrams illustrate that the steady states can appear or disappear only through saddle-node or transcritical bifurcations. Moreover, the operating diagrams describe the asymptotic behavior of this system by varying the control parameters and show the effect of the inhibition of predation on the emergence of the bistability region and the reduction until the disappearance of the coexistence region by increasing this inhibition parameter.

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Neimark-Sacker Bifurcation in Demand-Inventory Model with Stock-Level-Dependent Demand

Discrete Dynamics in Nature and Society ◽

10.1155/2017/8162865 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10

Author(s):

Piotr Hachuła ◽

Magdalena Nockowska-Rosiak ◽

Ewa Schmeidel

Keyword(s):

Equilibrium Points ◽

Three Dimensional ◽

Inventory Model ◽

Phase Portraits ◽

Dimensional System ◽

Planar System ◽

Stock Level ◽

Lyapunov Coefficient ◽

Analysis Of Dynamics ◽

Dependent Demand

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.

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Positive Solutions of a Diffusive Predator-Prey System including Disease for Prey and Equipped with Dirichlet Boundary Condition

Discrete Dynamics in Nature and Society ◽

10.1155/2016/2323752 ◽

2016 ◽

Vol 2016 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Xiaoqing Wen ◽

Yue Chen ◽

Hongwei Yin

Keyword(s):

Boundary Condition ◽

Positive Solutions ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Three Dimensional ◽

Disease Spread ◽

Dimensional System ◽

Predator Prey ◽

Predator Prey Model ◽

Fixed Point Index Theory

We study a three-dimensional system of a diffusive predator-prey model including disease spread for prey and with Dirichlet boundary condition and Michaelis-Menten functional response. By semigroup method, we are able to achieve existence of a global solution of this system. Extinction of this system is established by spectral method. By using bifurcation theory and fixed point index theory, we obtain existence and nonexistence of inhomogeneous positive solutions of this system in steady state.

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Phase-plane geometries in enzyme kinetics

Canadian Journal of Chemistry ◽

10.1139/v94-107 ◽

1994 ◽

Vol 72 (3) ◽

pp. 800-812 ◽

Cited By ~ 13

Author(s):

Simon J. Fraser ◽

Marc R. Roussel

Keyword(s):

Steady State ◽

Phase Plane ◽

Competitive Inhibition ◽

Three Dimensional ◽

Product Inhibition ◽

Slow Manifold ◽

Dimensional System ◽

Planar System ◽

State Behaviour ◽

Kinetics Of

The transient and steady-state behaviour of the reversible Michaelis–Menten mechanism [R] and Competitive Inhibition (CI) mechanism is studied by analysis in the phase plane. Usually, the kinetics of both mechanisms is simplified to give a modified Michaelis–Menten velocity expression; this applies to the CI mechanism with excess inhibitor and to mechanism [R] in the product inhibition limit. In this paper, [R] is treated exactly as a plane autonomous system of differential equations and its true (dynamical) steady state is described by a line-like slow manifold M. Initial velocity experiments for [R] no longer strictly correspond to the hyperbolic law (as in the irreversible Michaelis–Menten mechanism) and this leads to corrections to the standard integrated rate law. Using a new analysis, the slow dynamics of the CI mechanism is reduced from a three-dimensional system to a planar system. In this mechanism transient decay collapses the trajectory flow onto a two-dimensional "slow" surface Σ; motion on Σ can be treated exactly as projected dynamics in the plane. This projected flow may differ in important ways from that of two-step mechanisms, e.g., it may lack a proper steady state. The relevance of these more accurate dynamical descriptions is discussed in relation to experimental design and metabolic function.

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Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion

European Journal of Applied Mathematics ◽

10.1017/s0956792520000248 ◽

2020 ◽

pp. 1-31

Author(s):

HAI-YANG JIN ◽

ZHI-AN WANG

Keyword(s):

Pattern Formation ◽

Numerical Simulations ◽

Prey Density ◽

Temporal Patterns ◽

Steady States ◽

Global Dynamics ◽

Predator Prey ◽

Density Dependent ◽

Spatially Homogeneous ◽

Spatio Temporal

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) 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 parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

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Global properties of the three-dimensional predator-prey Lotka-Volterra systems

Journal of Applied Mathematics and Decision Sciences ◽

10.1155/s1173912699000085 ◽

1999 ◽

Vol 3 (2) ◽

pp. 155-162 ◽

Cited By ~ 22

Author(s):

A. Korobeinikov ◽

G. C. Wake

Keyword(s):

New Zealand ◽

Three Dimensional ◽

Indigenous Species ◽

Direct Lyapunov Method ◽

Predator Species ◽

Predator Prey ◽

Control Practice ◽

Global Properties ◽

Volterra Systems ◽

Pathological Case

The global properties of the classical three-dimensional Lotka-Volterra two prey-one predator and one prey-two predator systems, under the assumption that competition can be neglected, are analysed with the direct Lyapunov method. It is shown that, except for a pathological case, one species is always driven to extinction, and the system behaves asymptotically as a two-dimensional predator-prey Lotka-Volterra system. The same approach can be easily extended to systems with many prey species and one predator, or many predator species and one prey, and the same conclusion holds. The situation considered is common for New Zealand wild life, where indigenous and introduced species interact with devastating consequences for the indigenous species. According to our results the New Zealand indigenous species are definitely driven to extinction, not only in consequence of unsuccessful competition, but even when competition is absent. This result leads to a better understanding of the mechanism of natural selection, and gives a new insight into pest control practice.

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Convective instability in a diffusive predator–prey system

Electronic journal of qualitative theory of differential equations ◽

10.14232/ejqtde.2021.1.74 ◽

2021 ◽

pp. 1-9

Author(s):

Hui Chen ◽

Xuelian Xu

Keyword(s):

Steady State ◽

Neumann Boundary ◽

Reaction Diffusion ◽

Steady States ◽

Small Diffusion ◽

Predator Prey ◽

Positive Steady States ◽

Existence And Stability ◽

Biological Pattern Formation ◽

Turing Mechanism

It is well known that biological pattern formation is the Turing mechanism, in which a homogeneous steady state is destabilized by the addition of diffusion, though it is stable in the kinetic ODEs. However, steady states that are unstable in the kinetic ODEs are rarely mentioned. This paper concerns a reaction diffusion advection system under Neumann boundary conditions, where steady states that are unstable in the kinetic ODEs. Our results provide a stabilization strategy for the same steady state, the combination of large advection rate and small diffusion rate can stabilize the homogeneous equilibrium. Moreover, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis.

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Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species

Open Mathematics ◽

10.1515/math-2019-0082 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1186-1202 ◽

Cited By ~ 3

Author(s):

Fengde Chen ◽

Xinyu Guan ◽

Xiaoyan Huang ◽

Hang Deng

Keyword(s):

Birth Rate ◽

Allee Effect ◽

Prey Species ◽

Positive Equilibrium ◽

Volterra Type ◽

Predator Species ◽

Dynamic Behaviors ◽

Predator Prey ◽

Density Dependent ◽

Prey System

Abstract A Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species is proposed and studied. For non-delay case, such topics as the persistent of the system, the local stability property of the equilibria, the global stability of the positive equilibrium are investigated. For the system with infinite delay, by using the iterative method, a set of sufficient conditions which ensure the global attractivity of the positive equilibrium is obtained. By introducing the density dependent birth rate, the dynamic behaviors of the system becomes complicated. The system maybe collapse in the sense that both the species will be driven to extinction, or the two species could be coexist in a stable state. Numeric simulations are carried out to show the feasibility of the main results.

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