scholarly journals Partitioning a reaction–diffusion ecological network for dynamic stability

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180524
Author(s):  
Dinesh Kumar ◽  
Jatin Gupta ◽  
Soumyendu Raha
Keyword(s):  
Reaction Diffusion ◽  
Ecological Network ◽  
Graph Models ◽  
Predator Prey ◽  
Habitat Patches ◽  
Bisection Algorithm ◽  
Communication Links ◽  
Partitioned Networks ◽  
Partitioning Procedure ◽  
The Stability

The loss of dispersal connections between habitat patches may destabilize populations in a patched ecological network. This work studies the stability of populations when one or more communication links is removed. An example is finding the alignment of a highway through a patched forest containing a network of metapopulations in the patches. This problem is modelled as that of finding a stable cut of the graph induced by the metapopulations network, where nodes represent the habitat patches and the weighted edges model the dispersal between habitat patches. A reaction–diffusion system on the graph models the dynamics of the predator–prey system over the patched ecological network. The graph Laplacian's Fiedler value, which indicates the well-connectedness of the graph, is shown to affect the stability of the metapopulations. We show that, when the Fiedler value is sufficiently large, the removal of edges without destabilizing the dynamics of the network is possible. We give an exhaustive graph partitioning procedure, which is suitable for smaller networks and uses the criterion for both the local and global stability of populations in partitioned networks. A heuristic graph bisection algorithm that preserves the preassigned lower bound for the Fiedler value is proposed for larger networks and is illustrated with examples.

Pattern Formation Controlled by External Forcing in a Spatial Harvesting Predator-Prey Model

2011 ◽  
pp. 214-221
Author(s):  
Feng Rao
Keyword(s):  
Reaction Diffusion ◽  
Euler Method ◽  
External Forcing ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
Predator Prey Models ◽  
The Stability

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

A Semi-Analytical Method for Solving Problems on the Role of Prey Taxis in a Biological Control-Mathematical Model

2019 ◽  
Vol 10 (02) ◽  
pp. 1850009
Author(s):  
OPhir Nave ◽  
Yifat Baron ◽  
Manju Sharma
Keyword(s):  
Biological Control ◽  
Analytical Method ◽  
Reaction Diffusion ◽  
Prey Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Homotopy Analysis ◽  
The Stability ◽  
Pest Density

In this paper, we applied the well-known homotopy analysis methods (HAM), which is a semi-analytical method, perturbation method, to study a reaction–diffusion–advection model for the dynamics of populations under biological control. According to the predator–prey model, the advection expression represents the predator density movement in which the acceleration is proportional to the prey density gradient. The prey population reproduces logistically, and the interactions of prey population obey the Holling’s prey-dependent Type II functional response. The predation process splits into the following subdivided processes: random movement which is represented by diffusion, direct movement which is described by prey taxis, local prey interactions, and consumptions which are represented by the trophic function. In order to ensure a successful biological control, one should make the predator-pest population to stabilize at a very low level of pest density. One reason for this effect is the intermediate taxis activity. However, when the system loses stability, for example very intensive prey taxis destroys the stability, it leads to chaotic dynamics with pronounced outbreaks of pest density.

STABILITY AND HOPF BIFURCATION FOR A THREE-SPECIES REACTION–DIFFUSION PREDATOR–PREY SYSTEM WITH TWO DELAYS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350194
Author(s):  
GAO-XIANG YANG ◽  
JIAN XU
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Bifurcating Periodic Solution

In this paper, a three-species predator–prey system with diffusion and two delays is investigated. By taking the sum of two delays as a bifurcation parameter, it is found that the spatially homogeneous Hopf bifurcation can occur as the sum of two delays crosses a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are obtained by employing the center manifold theorem and the normal form theory. In addition, some numerical simulations are also given to illustrate the theoretical analysis.

scholarly journals Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1244
Author(s):  
Maria Francesca Carfora ◽  
Isabella Torcicollo
Keyword(s):  
Stability Analysis ◽  
Reaction Diffusion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Fear Effect ◽  
Group Defense ◽  
The Cross ◽  
The Stability

In this paper, a reaction-diffusion prey-predator system including the fear effect of predator on prey population and group defense has been considered. The conditions for the onset of cross-diffusion-driven instability are obtained by linear stability analysis. The technique of multiple time scales is employed to deduce the amplitude equation near Turing bifurcation threshold by choosing the cross-diffusion coefficient as a bifurcation parameter. The stability analysis of these amplitude equations leads to the identification of various Turing patterns driven by the cross-diffusion, which are also investigated through numerical simulations.

Stationary patterns for a Leslie–Gower type three species model with diffusion

2014 ◽  
Vol 07 (06) ◽  
pp. 1450069 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu
Keyword(s):  
Global Stability ◽  
Positive Constant ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Steady States ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Positive Steady States ◽  
The Stability

In this paper, a diffusive three species predator–prey model with two Leslie–Gower terms is considered. The stability of the unique positive constant equilibrium for the reaction–diffusion system is obtained. Sufficient conditions are derived for the global stability of the positive constant equilibrium. In particular, we establish the existence and non-existence of non-constant positive steady states of this system. The results indicate that the large diffusivity is helpful for the appearance of the non-constant positive steady states (stationary patterns).

scholarly journals Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Delayed Reaction ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
The Stability

A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.

scholarly journals Inhomogeneous spatial patterns in diffusive predator-prey system with spatial memory and predator-taxis

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Spatial Memory ◽  
Reaction Diffusion ◽  
Diffusion Term ◽  
Predator Prey ◽  
Spatially Inhomogeneous ◽  
Prey System ◽  
Mode 2 ◽  
The Stability

Abstract In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.

scholarly journals POSITIVE SOLUTIONS BIFURCATING FROM ZERO SOLUTION IN A PREDATOR-PREY REACTION–DIFFUSION SYSTEM

2011 ◽  
Vol 16 (4) ◽  
pp. 558-568 ◽  
Author(s):  
Cun-Hua Zhang ◽  
Xiang-Ping Yan
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Reduction Method ◽  
Reaction Diffusion ◽  
Zero Solution ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Predator Prey ◽  
The Stability

An elliptic system subject to the homogeneous Dirichlet boundary con- dition denoting the steady-state system of a two-species predator-prey reaction– diffusion system with the modified Leslie–Gower and Holling-type II schemes is con- sidered. By using the Lyapunov–Schmidt reduction method, the bifurcation of the positive solution from the trivial solution is demonstrated and the approximated ex- pressions of the positive solutions around the bifurcation point are also given accord- ing to the implicit function theorem. Finally, by applying the linearized method, the stability of the bifurcating positive solution is also investigated. The results obtained in the present paper improved the existing ones.

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