scholarly journals Coexistence for an Almost Periodic Predator-Prey Model with Intermittent Predation Driven by Discontinuous Prey Dispersal

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Yantao Luo ◽  
Long Zhang ◽  
Zhidong Teng ◽  
Tingting Zheng
Keyword(s):  
Growth Rates ◽  
Prey Species ◽  
Impulsive Differential Equations ◽  
Comparison Theorems ◽  
Almost Periodic ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models

An almost periodic predator-prey model with intermittent predation and prey discontinuous dispersal is studied in this paper, which differs from the classical continuous and impulsive dispersal predator-prey models. The intermittent predation behavior of the predator species only happens in the channels between two patches where the discontinuous migration movement of the prey species occurs. Using analytic approaches and comparison theorems of the impulsive differential equations, sufficient criteria on the boundedness, permanence, and coexistence for this system are established. Finally, numerical simulations demonstrate that, for an intermittent predator-prey model, both the intermittent predation and intrinsic growth rates of the prey and predator species can greatly impact the permanence, extinction, and coexistence of the population.

The fundamentals of predator–prey interactions

2019 ◽  
pp. 81-95
Author(s):  
Gary G. Mittelbach ◽  
Brian J. McGill
Keyword(s):  
Steady State ◽  
Species Interactions ◽  
Predator Species ◽  
Trade Off ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fundamental Building Block ◽  
Prey Model ◽  
Predator Prey Models ◽  
Consumer Resource

This chapter introduces the concept of the consumer-resource link, the idea that each species in a community consumes resources and is itself consumed by other species. The consumer–resource link is the fundamental building block from which more-complex food chains and food webs are constructed. The chapter continues by exploring what is arguably the simplest consumer–resource interaction—one predator species feeding on one species of prey. Important topics discussed in the context of predator–prey interactions are the predator’s functional response, the Lotka–Volterra predator–prey model, the Rosenzweig–MacArthur predator–prey model, and the suppression-stability trade-off. Isocline analysis is introduced as a method for visualizing the outcome of species interactions at steady-state or equilibrium. Herbivory and parasitism are briefly discussed within the context of general predator–prey models.

RATIO-DEPENDENT PREDATOR–PREY MODEL WITH STAGE STRUCTURE AND TIME DELAY

2012 ◽  
Vol 05 (04) ◽  
pp. 1250014 ◽  
Author(s):  
LIJUAN ZHA ◽  
JING-AN CUI ◽  
XUEYONG ZHOU
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent ◽  
Predator Prey Models

Ratio-dependent predator–prey models are favored by many animal ecologists recently as more suitable ones for predator–prey interactions where predation involves searching process. In this paper, a ratio-dependent predator–prey model with stage structure and time delay for prey is proposed and analyzed. In this model, we only consider the stage structure of immature and mature prey species and not consider the stage structure of predator species. We assume that the predator only feed on the mature prey and the time for prey from birth to maturity represented by a constant time delay. At first, we investigate the permanence and existence of the proposed model and sufficient conditions are derived. Then the global stability of the nonnegative equilibria are derived. We also get the sufficient criteria for stability switch of the positive equilibrium. Finally, some numerical simulations are carried out for supporting the analytic results.

GLOBAL ANALYSIS OF A HARVESTED PREDATOR–PREY MODEL INCORPORATING A CONSTANT PREY REFUGE

2010 ◽  
Vol 03 (02) ◽  
pp. 205-223 ◽  
Author(s):  
LIUJUAN CHEN ◽  
FENGDE CHEN
Keyword(s):  
Prey Species ◽  
Global Analysis ◽  
Sufficient Conditions ◽  
Equilibrium Density ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Density Values

A predator–prey model with Holling type II functional response incorporating a constant prey refuge and independent harvesting in either species is investigated. Some sufficient conditions of the instability and stability properties to the equilibria and the existence and uniqueness to limit cycles for the model are obtained. We also show that influence of prey refuge and harvesting efforts on equilibrium density values. One of the surprising finding is that for fixed prey refuge, harvesting has no influence on the final density of the prey species, while the density of predator species is decreasing with the increasing of harvesting effort on prey species and the fixation of harvesting effort on predator species. Numerical simulations are carried out to illustrate the obtained results and the dependence of the dynamic behavior on the harvesting efforts or prey refuge.

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.

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

scholarly journals Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete Predator-Prey Models in Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Chunqing Wu ◽  
Shengming Fan ◽  
Patricia J. Y. Wong
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Studies ◽  
Patchy Environment ◽  
Predator Prey ◽  
Numerical Examples ◽  
Predator Prey Model ◽  
Prey Model ◽  
Dispersal Corridors ◽  
Predator Prey Models ◽  
Theoretical Results

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

Almost periodic solution of a modified Leslie–Gower predator–prey model with Holling-type II schemes and mutual interference

2014 ◽  
Vol 07 (03) ◽  
pp. 1450028 ◽  
Author(s):  
Shengbin Yu ◽  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
Mutual Interference ◽  
Almost Periodic Solution ◽  
Type Ii ◽  
Almost Periodic ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Globally Attractive

In this paper, we consider a modified Leslie–Gower predator–prey model with Holling-type II schemes and mutual interference. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov function, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. Our results not only supplement but also improve some existing ones.

scholarly journals Global dynamics and parameter identifiability in a predator-prey interaction model

2018 ◽  
Vol 5 (1) ◽  
pp. 113-126
Author(s):  
Jai Prakash Tripathi ◽  
Suraj S. Meghwani ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Interaction Model ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Permanence Property

AbstractThis paper discusses a predator-prey model with prey refuge. We investigate the role of prey refuge on the existence and stability of the positive equilibrium. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional, which shows that the prey refuge has no influence on the permanence property of the system. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. To access the usability of proposed predator-prey model in practical scenarios, we also suggest, the use of Levenberg-Marquardt (LM) method for associated parameter estimation problem. Numerical results demonstrate faithful reconstruction of system dynamics by estimated parameter by LM method. The analytical results found in this paper are illustrated with the help of suitable numerical examples

DYNAMICS OF MODIFIED PREDATOR-PREY MODELS

2010 ◽  
Vol 20 (09) ◽  
pp. 2657-2669 ◽  
Author(s):  
P. E. KLOEDEN ◽  
C. PÖTZSCHE
Keyword(s):  
Global Attractor ◽  
Continuous Time ◽  
Explicit Representation ◽  
Mass Action ◽  
Volterra Equations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Large Populations ◽  
Predator Prey Models

Besides being structurally unstable, the Lotka–Volterra predator-prey model has another shortcoming due to the invalidity of the principle of mass action when the populations are very small. This leads to extremely large populations recovering from unrealistically small ones. The effects of linear modifications to structurally unstable continuous-time predator-prey models in a (small) neighbourhood of the origin are investigated here. In particular, it is shown that typically either a global attractor or repeller arises depending on the choice of coefficients.The analysis is based on Poincaré mappings, which allow an explicit representation for the classical Lotka–Volterra equations.

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