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.

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 The Study of a Predator-Prey Model with Fear Effect Based on State-Dependent Harvesting Strategy

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-19
Author(s):  
Y. Tian ◽  
H. M. Li
Keyword(s):  
Numerical Simulation ◽  
Periodic Solution ◽  
Prey Population ◽  
Global Dynamics ◽  
Predator Population ◽  
Positive Order ◽  
Predator Prey ◽  
Predator Prey Model ◽  
State Dependent ◽  
The Stability

In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- q ( q ≥ 1 ) periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- q periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB.

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 Dynamics Analysis of Modified Leslie-Gower Model with Simplified Holling Type IV Functional Response

2021 ◽  
Vol 18 (1) ◽  
pp. 12-21
Author(s):  
Nur Suci Ramadhani ◽  
Toaha Toaha ◽  
Kasbawati Kasbawati
Keyword(s):  
Population Density ◽  
Equilibrium Point ◽  
Functional Response ◽  
Prey Population ◽  
Dynamics Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Simulation Results ◽  
The Stability

In this paper, the modified Leslie-Gower predator-prey model with simplified Holling type IV functional response is discussed. It is assumed that the prey population is a dangerous population. The equilibrium point of the model and the stability of the coexistence equilibrium point are analyzed. The simulation results show that both prey and predator populations will not become extinct as time increases. When the prey population density increases, there is a decrease in the predatory population density because the dangerous prey population has a better ability to defend itself from predators when the number is large enough.

EFFECT OF PARASITIC INFECTION IN THE LESLIE–GOWER PREDATOR–PREY MODEL

2008 ◽  
Vol 16 (03) ◽  
pp. 425-444 ◽  
Author(s):  
MAINUL HAQUE ◽  
EZIO VENTURINO
Keyword(s):  
Biological Control ◽  
Control Parameter ◽  
Disease Incidence ◽  
Parasitic Infection ◽  
Logistic Growth ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

The Leslie–Gower predator–prey model with logistic growth in prey is here modified to include an SI parasitic infection affecting the prey population only. Thresholds are identified for the predator population to survive, and the conditions for the disease to die out naturally are given. The behavior of the system around each equilibrium is investigated, showing that the disease incidence may have a relevant influence on the dynamics of complex ecosytems, assuming at times the role of a biological control parameter.

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 Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Shengmao Fu ◽  
Lina Zhang
Keyword(s):  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Sex Structure ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Ode System ◽  
Prey Model ◽  
Weakly Coupled ◽  
Coupled Reaction

In this paper, we consider a cross-diffusion predator-prey model with sex structure. We prove that cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system. As a result, we find that stationary patterns arise solely from the effect of cross-diffusion.

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