ALLEE EFFECT, EMIGRATION AND IMMIGRATION IN A CLASS OF PREDATOR-PREY MODELS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 195-215 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JAIME MENA-LORCA ◽  
HÉCTOR MENESES-ALCAY ◽  
BETSABÉ GONZÁLEZ-YAÑEZ ◽  
JOSÉ D. FLORES
Keyword(s):  
Allee Effect ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Threshold Level ◽  
Prey Population ◽  
Stable Limit ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stable Attractor ◽  
Predator Prey Models

In this work we analyze a predator-prey model proposed by A. Kent et al. in Ecol. Model.162, 233 (2003), in which two aspect of the model are considered: an effect of emigration or immigration on prey population to constant rate and a prey threshold level for predators. We prove that the system when the immigration effect is introduced in the model has a dynamics that is similar to the Rosenzweig-MacArthur model. Also, when emigration is considered in the model, we show that the behavior of the system is strongly dependent on this phenomenon, this due to the fact that trajectories are highly sensitive to the initial conditions, in similar way as when Allee effect is assumed on prey. Furthermore, we determine constraints in the parameters space for which two stable attractor exist, indicating that the extinction of both population is possible in addition with the coexistence of oscillating of populations size in a unique stable limit cycle. We also show that the consideration of a threshold level of prey population for the predator is not essential in the dynamics of the model.

scholarly journals Bifurcation analysis of the predator–prey model with the Allee effect in the predator

2021 ◽  
Vol 84 (1-2) ◽  
Author(s):  
Deeptajyoti Sen ◽  
Saktipada Ghorai ◽  
Malay Banerjee ◽  
Andrew Morozov
Keyword(s):  
Allee Effect ◽  
Mathematical Formulation ◽  
Global Bifurcation ◽  
Predator Population ◽  
Stable Limit ◽  
Bifurcation Structure ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models

AbstractThe use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2016 ◽  
Vol 26 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Jicai Huang ◽  
Xiaojing Xia ◽  
Xinan Zhang ◽  
Shigui Ruan
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Positive Equilibrium ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Stable Equilibria

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amit K. Pal
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonlinear Harvesting

Abstract In this paper, the dynamical behaviors of a delayed predator–prey model (PPM) with nonlinear harvesting efforts by using imprecise biological parameters are studied. A method is proposed to handle these imprecise parameters by using a parametric form of interval numbers. The proposed PPM is presented with Crowley–Martin type of predation and Michaelis–Menten type prey harvesting. The existence of various equilibrium points and the stability of the system at these equilibrium points are investigated. Analytical study reveals that the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate the main analytical findings.

DETERMINISTIC PREDATOR–PREY MODELS WITH DISEASE IN THE PREY POPULATION

2020 ◽  
Vol 28 (03) ◽  
pp. 751-784
Author(s):  
SOPHIA R.-J. JANG ◽  
HSIU-CHUAN WEI
Keyword(s):  
Initial Conditions ◽  
Sufficient Conditions ◽  
Host Population ◽  
Trophic Levels ◽  
Prey Population ◽  
Model Parameters ◽  
Uniform Persistence ◽  
Predator Prey ◽  
Predator Prey Models ◽  
Infected Prey

A class of deterministic predator–prey systems, where the prey population is subject to an infectious disease, is studied. The disease can be transmitted both horizontally and vertically within the host population but cannot be spread between the two trophic levels. Using the mathematical tools of uniform persistence, we derive sufficient conditions for which the interacting populations can coexist. Criteria based on model parameters for which either only the infected prey or healthy prey persists are also provided. It is found through numerical investigations that predation can change competition outcomes between healthy and infected prey populations. Depending on the parameter regimes and initial conditions, predation can either eradicate or promote the disease. As infected prey provides a resource for the predators, disease may promote persistence of the predators. However, infected prey may dominate the predator–prey community if disease is very infectious. In addition, disease in the prey population can mediate a hydra effect in predators and may induce chaotic interactions.

scholarly journals Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Zongmin Yue ◽  
Xiaoqin Wang ◽  
Haifeng Liu
Keyword(s):  
Allee Effect ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Highly Sensitive ◽  
The Impact

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

scholarly journals A predator-prey model with allee effect and fast strategy evolution dynamics of predators using hawk and dove tactics

2011 ◽  
Vol 50 (1) ◽  
pp. 13-24
Author(s):  
Jitka Kühnová ◽  
Lenka Přibylová
Keyword(s):  
Limit Cycles ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Prey Population ◽  
Bifurcation Of Limit Cycles ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Bifurcations ◽  
Prey Model ◽  
Evolution Dynamics

ABSTRACT In this work we present the predator-prey model with Allee effect and Hawk and Dove tactics in fighting over caught prey implemented as fast strategy evolution dynamics. We extend the work of Auger, Parra, Morand and S´anchez (2002) using the prey population embodying Allee effect and analogously to this work we get two connected submodels with polymorphic and monomorphic predator population.We get much richer dynamics, in each submodel we find local bifurcations (saddle-node, supercritical Hopf caused by Allee effect and Bogdanov- -Takens) and a global bifurcation of limit cycles caused by the strategy evolution that is not possible in any of the submodels that can lead to a bluesky extinction of both populations.

A delayed predator–prey model with strong Allee effect in prey population growth

2011 ◽  
Vol 68 (1-2) ◽  
pp. 23-42 ◽  
Author(s):  
Pallav Jyoti Pal ◽  
Tapan Saha ◽  
Moitri Sen ◽  
Malay Banerjee
Keyword(s):  
Population Growth ◽  
Allee Effect ◽  
Prey Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Strong Allee Effect

scholarly journals Density Dependent Delayed Migration for Rosenzweig-Macaurther Model with Holling Type II Predator Functional Response

2019 ◽  
pp. 1-12
Author(s):  
Samuel B. Apima ◽  
George O. Lawi ◽  
Nthiiri J. Kagendo
Keyword(s):  
Time Delay ◽  
Prey Species ◽  
Prey Population ◽  
Population Increase ◽  
Logistic Growth ◽  
Natural Habitats ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Predator Prey Models

The model describing the interaction between the predator and prey species is referred to as a predator-prey model. The migration of these species from one patch to another may not be instantaneous. This may be due to barriers such as a swollen river or a busy infrastructure through the natural habitat. Recent predator-prey models have either incorporated a logistic growth for the prey population or a time delay in migration of the two species. Predator-prey models with logistic growth that integrate time delays in density-dependent migration of both species have been given little attention. A Rosenzweig-MacAurther model with density-dependent migration and time delay in the migration of both species is developed and analyzed in this study. The Analysis of the model when the prey migration rate is greater than or equal to the prey growth rate, the two species will coexist, otherwise, at least one species will become extinct. A longer delay slows down the rate at which the predator and prey population increase or decrease, thus aecting the population density of these species. The prey migration due to the predator density does not greatly affect the prey density and existence compared to the other factors that cause the prey to migrate. These factors include human activities in the natural habitats like logging and natural causes like bad climatic conditions, limited food resources and overpopulation of the prey species in a patch among others.

scholarly journals Erratum to: A delayed predator–prey model with strong Allee effect in prey population growth

2012 ◽  
Vol 68 (1-2) ◽  
pp. 43-43
Author(s):  
Pallav Jyoti Pal ◽  
Tapan Saha ◽  
Moitri Sen ◽  
Malay Banerjee
Keyword(s):  
Population Growth ◽  
Allee Effect ◽  
Prey Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Strong Allee Effect

Qualitative properties and bifurcations of discrete-time Bazykin–Berezovskaya predator–prey model

2020 ◽  
Vol 13 (06) ◽  
pp. 2050040
Author(s):  
A. A. Elsadany ◽  
Qamar Din ◽  
S. M. Salman
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Initial Conditions ◽  
Qualitative Properties ◽  
Step Size ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model ◽  
Predator Prey Model ◽  
Prey Model

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

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