scholarly journals Coexistence of the Three Trophic Levels in a Model with Intraguild Predation and Intraspecific Competition of Prey

2021 ◽  
Vol 16 (2) ◽  
pp. 394-410
Author(s):  
Evgeniya Giricheva
Keyword(s):  
Intraspecific Competition ◽  
Intraguild Predation ◽  
Local Stability ◽  
Consumption Rate ◽  
Present Model ◽  
Trophic Levels ◽  
Positive Equilibrium ◽  
Functional Responses ◽  
Equilibrium Coexistence ◽  
Existence And Stability

The model of a three-trophic community with intraguild predation is considered. The system consists of three coupled ordinary differential equations describing the dynamics of resource, prey and predator. Models with intraguild predation are characterized by predators that feed on resource of its own prey. A number of similar models with different functional responses have been proposed. In contrast to previous works, in the present model, the predator functional response to the resource is differed from that to the prey. The model takes into account an intraspecific competition of prey to stabilize the system in resource-rich environment. Conditions of existence and local stability of non-negative solutions are established. The possibility of Hopf bifurcation around positive equilibrium with consumption rate as bifurcation parameter is studied. For the model, in the plane of the consumption and predation rates, the regions of existence and stability of boundary and internal equilibria are constructed. Numerical simulations show that the region of equilibrium coexistence of populations is increased due to the inclusion of prey self-limitation in the model. Bifurcation diagrams confirm the stabilizing effect of intraspecific competition of prey on the system dynamics in resource-rich environment.

Qualitative Analysis of Three-Dimensional Single Food-Chain Model with Variable Consumption Rate in Chemostat

2014 ◽  
Vol 955-959 ◽  
pp. 463-470
Author(s):  
Jing Liu ◽  
Hong Wei Jiang ◽  
Chao Liu
Keyword(s):  
Food Chain ◽  
Consumption Rate ◽  
Three Dimensional ◽  
Quadratic Function ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Equilibrium Solutions ◽  
Existence And Stability ◽  
Variable Consumption ◽  
Food Chain Model

The paper studies three-dimensional food-chain model with variable consumption rate in Chemostat. Assume the prey population's consumption rate of the nutrients is quadratic function, and the predator's consumption rate of the prey population is linear function. Use qualitative theory of ordinary differential equation to analyze the equilibrium solution of the model, especially the existence and stability of positive equilibrium solutions and Hopf bifurcation solutions. Finally,several numerical simulations illustrating the theoretical analysis are also given.

scholarly journals Global Stability of a Predator-Prey Model with Ivlev-Type Functional Response

2020 ◽  
Vol 99 (99) ◽  
pp. 1-12
Author(s):  
Yinshu Wu ◽  
Wenzhang Huang
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models ◽  
Global And Local

A predator-prey model with Ivlev-Type functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional responses.

Global dynamics of delayed intraguild predation model with intraspecific competition

2018 ◽  
Vol 11 (08) ◽  
pp. 1850116
Author(s):  
Zhenzhen Li ◽  
Binxiang Dai
Keyword(s):  
Periodic Solution ◽  
Global Existence ◽  
Intraspecific Competition ◽  
Intraguild Predation ◽  
Critical Value ◽  
Existence Results ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
The Stability ◽  
Theoretical Results

A delayed intraguild predation (IGP) model with intraspecific competition is considered. It is shown that the delay has a destabilizing effect and induces oscillations. The global existence results of periodic solutions bifurcating from the positive equilibrium are established. It is shown that there exists at least one nontrival periodic solution when the delay passes through a certain critical value. Numerical simulations are performed to illustrate our theoretical results and show that intraspecific competition can also affect the stability of the positive equilibrium of the system.

Practical persistence in ecological models via comparison methods

1996 ◽  
Vol 126 (2) ◽  
pp. 247-272 ◽  
Author(s):  
Robert Stephen Cantrell ◽  
Chris Cosner
Keyword(s):  
Reaction Diffusion ◽  
Basic Question ◽  
Positive Equilibrium ◽  
Uniform Persistence ◽  
Interacting Species ◽  
Attracting Set ◽  
Alternative Approach ◽  
Interacting Populations ◽  
Existence And Stability ◽  
Stable Periodic

A basic question in mathematical ecology is that of deciding whether or not a model for the population dynamics of interacting species predicts their long-term coexistence. A sufficient condition for coexistence is the presence of a globally attracting positive equilibrium, but that condition may be too strong since it excludes other possibilities such as stable periodic solutions. Even if there is such an equilibrium, it may be difficult to establish its existence and stability, especially in the case of models with diffusion. In recent years, there has been considerable interest in the idea of uniform persistence or permanence, where coexistence is inferred from the existence of a globally attracting positive set. The advantage of that approach is that often uniform persistence can be shown much more easily than the existence of a globally attracting equilibrium. The disadvantage is that most techniques for establishing uniform persistence do not provide any information on the size or location of the attracting set. That is a serious drawback from the applied viewpoint, because if the positive attracting set contains points that represent less than one individual of some species, then the practical interpretation that uniform persistence predicts coexistence may not be valid. An alternative approach is to seek asymptotic lower bounds on the populations or densities in the model, via comparison with simpler equations whose dynamics are better known. If such bounds can be obtained and approximately computed, then the prediction ofpersistence can be made practical rather than merely theoretical. This paper describes how practical persistence can be established for some classes of reaction–diffusion models for interacting populations. Somewhat surprisingly, themodels need not be autonomous or have any specific monotonicity properties.

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

scholarly journals Cascading effects of a top predator on intraspecific competition at intermediate and basal trophic levels

2018 ◽  
Vol 32 (9) ◽  
pp. 2241-2252 ◽  
Author(s):  
Catherine M. Matassa ◽  
Patrick J. Ewanchuk ◽  
Geoffrey C. Trussell
Keyword(s):  
Intraspecific Competition ◽  
Trophic Levels ◽  
Top Predator ◽  
Cascading Effects

The Community of Ecological Opportunities

2017 ◽  
Author(s):  
Mark A. McPeek
Keyword(s):  
Plant Species ◽  
Resource Availability ◽  
Intraguild Predation ◽  
Species Interactions ◽  
Apparent Competition ◽  
Predation Pressure ◽  
Trophic Levels ◽  
Model Species ◽  
Pollinator Species ◽  
Different Types

This chapter examines ecological opportunities that are available to species in various positions within a biological community, with particular emphasis on identifying the criteria necessary for an ecological opportunity to exist. Before discussing what performance capabilities a species must have to fill different types of ecological opportunities and what is required for invasibility of species into different functional positions in a community, the chapter considers the different frameworks that have been used to model species interactions. It then describes resource and apparent competition to show how resource availability from below and predation pressure from above can affect the types of species that can exploit specifc ecological opportunities. It also analyzes communities with three trophic levels, intraguild predation or omnivory, mutualism, the mechanisms that foster coexistence between one plant species and one pollinator species, and the case of one plant species with multiple pollinators.

scholarly journals A Novel Method to Determine the Local Stability of the n-Species Lotka-Volterra System with Multiple Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Xiao-Ping Chen ◽  
Hao Liu
Keyword(s):  
Asymptotic Stability ◽  
Local Stability ◽  
Integral Method ◽  
Numerical Experiments ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Volterra System ◽  
Local Asymptotic Stability ◽  
Discrete Delays ◽  
Novel Method

The n-species Lotka-Volterra system with discrete delays is considered. The local asymptotic stability of positive equilibrium is investigated based on a contour integral method. The main purpose of this paper is to propose a new and general algorithm to study the local asymptotic stability of the positive equilibrium for then-dimensional Lotka-Volterra system. Some numerical experiments are carried out to show the effectiveness of the proposed method.

Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays

2013 ◽  
Vol 756-759 ◽  
pp. 2857-2862
Author(s):  
Shun Yi Li ◽  
Wen Wu Liu
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Characteristic Equations ◽  
Numerical Examples ◽  
Predator Model ◽  
Predator System

A three-stage-structured prey-predator model with multi-delays is considered. The characteristic equations and local stability of the equilibrium are analyzed, and the conditions for the positive equilibrium occurring Hopf bifurcation are obtained by applying the theorem of Hopf bifurcation. Finally, numerical examples and brief conclusion are given.

QUALITATIVE ANALYSIS FOR THE MERKIN ENZYME REACTION SYSTEM

2002 ◽  
Vol 10 (02) ◽  
pp. 167-182
Author(s):  
YUQUAN WANG ◽  
ZUORUI SHEN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Reaction System ◽  
Enzyme Reaction ◽  
Qualitative Theory ◽  
Positive Equilibrium ◽  
Hopf Bifurcations

Applying qualitative theory and Hopf bifurcation theory, we detailedly discuss the Merkin enzyme reaction system, and the sufficient conditions derived for the global stability of the unique positive equilibrium, the local stability of three equilibria and the existence of limit cycles. Meanwhile, we show that the Hopf bifurcations may occur. Using MATLAB software, we present three examples to simulate these conclusions in this paper.

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