scholarly journals Dynamics of a Predator-Prey System with Mixed Functional Responses

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hunki Baek ◽  
Dongseok Kim
Keyword(s):  
Equilibrium Points ◽  
Functional Responses ◽  
Predator Prey ◽  
Numerical Examples ◽  
Persistence Conditions ◽  
Prey System ◽  
Different Types ◽  
Theoretical Results

A predator-prey system with two preys and one predator is considered. Especially, two different types of functional responses, Holling type and Beddington-DeAngelis type, are adopted. First, the boundedness of system is showed. Stabilities analysis of system is investigated via some properties about equilibrium points and stabilities of two subsystems without one of the preys of system. Also, persistence conditions of system are found out and some numerical examples are illustrated to substantiate our theoretical results.

scholarly journals Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

2008 ◽  
Vol 2008 ◽  
pp. 1-15 ◽  
Author(s):  
Can-Yun Huang ◽  
Min Zhao ◽  
Hai-Feng Huo
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Ii Functional Response

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

scholarly journals Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Kankan Sarkar ◽  
Subhas Khajanchi ◽  
Prakash Chandra Mali ◽  
Juan J. Nieto
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Prey Population ◽  
Uniform Persistence ◽  
Functional Responses ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

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.

Bogdanov–Takens bifurcation with codimension three of a predator–prey system suffering the additive Allee effect

2017 ◽  
Vol 10 (03) ◽  
pp. 1750044 ◽  
Author(s):  
Yanwei Liu ◽  
Zengrong Liu ◽  
Ruiqi Wang
Keyword(s):  
Allee Effect ◽  
Perturbation Parameter ◽  
Parameter Plane ◽  
Predator Prey ◽  
Numerical Examples ◽  
Codimension Two ◽  
Prey System ◽  
Existence Conditions ◽  
Parameter Dependent

In the present work, research efforts have focused on investigating codimension two and three Bogdanov–Takens bifurcations of a predator–prey system with additive Allee effect. According to the existence conditions of Bogdanov–Takens bifurcation, we give the associated generic unfolding, and derive the dynamical classification in the perturbation parameter plane using some smooth parameter-dependent transformations of coordinate. Moreover, some numerical examples and simulations are performed to complete and illustrate our results.

scholarly journals Space race functional responses

2015 ◽  
Vol 282 (1801) ◽  
pp. 20142121 ◽  
Author(s):  
Henrik Sjödin ◽  
Åke Brännström ◽  
Göran Englund
Keyword(s):  
Functional Response ◽  
Community Dynamics ◽  
Simple Structure ◽  
Functional Responses ◽  
Response Models ◽  
Predator Prey ◽  
Prey System ◽  
Space Race ◽  
Functional Response Models ◽  
The Relationship

We derive functional responses under the assumption that predators and prey are engaged in a space race in which prey avoid patches with many predators and predators avoid patches with few or no prey. The resulting functional response models have a simple structure and include functions describing how the emigration of prey and predators depend on interspecific densities. As such, they provide a link between dispersal behaviours and community dynamics. The derived functional response is general but is here modelled in accordance with empirically documented emigration responses. We find that the prey emigration response to predators has stabilizing effects similar to that of the DeAngelis–Beddington functional response, and that the predator emigration response to prey has destabilizing effects similar to that of the Holling type II response. A stability criterion describing the net effect of the two emigration responses on a Lotka–Volterra predator–prey system is presented. The winner of the space race (i.e. whether predators or prey are favoured) is determined by the relationship between the slopes of the species' emigration responses. It is predicted that predators win the space race in poor habitats, where predator and prey densities are low, and that prey are more successful in richer habitats.

scholarly journals Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Xinyou Meng ◽  
Qingling Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Complex Dynamics ◽  
Hybrid Control ◽  
Positive Equilibrium ◽  
Stochastic Fluctuation ◽  
Predator Prey ◽  
Prey System ◽  
Singularity Induced Bifurcation ◽  
Distribution Of Roots ◽  
Theoretical Results

A singular delayed biological economic predator-prey system with and without stochastic fluctuation is proposed. The conditions of singularity induced bifurcation are gained, and a state feedback controller is designed to eliminate such bifurcation. Furthermore, saddle-node bifurcation is also showed. Next, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of roots of the corresponding characteristic equation, and the hybrid control strategy is used to control the occurrence of Hopf bifurcation. In addition, some explicit formulas determining the spectral densities of the populations and harvest effort are given when the system is considered with stochastic fluctuation. Finally, numerical simulations are illustrated to verify the theoretical results.

scholarly journals Dynamics of a Second-Order System of Nonlinear Difference Equations

2019 ◽  
Author(s):  
Erkan Taşdemir
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Equilibrium Points ◽  
Order System ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Numerical Examples ◽  
Unbounded Solutions ◽  
Theoretical Results

In this paper, we investigate the equilibrium points, stability of two equilibrium points, convergences of negative equilibrium point, periodic solutions, and existence of bounded or unbounded solutions of a system of nonlinear difference equations xn+1 =xn-1yn - 1, yn+1 = yn-1xn - 1 n = 0,1,..., where the initial values are real numbers. Additionally we present some numerical examples to verify our theoretical results.

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

scholarly journals Predator-prey nutrient competition undermines predator coexistence

2019 ◽  
Author(s):  
Toni Klauschies ◽  
Ursula Gaedke
Keyword(s):  
Nutrient Limitation ◽  
Uptake Rate ◽  
Species Coexistence ◽  
Logistic Growth ◽  
Nutrient Competition ◽  
Functional Responses ◽  
Predator Prey ◽  
Natural Systems ◽  
Prey System ◽  
Flow Through

AbstractContemporary theory of predator coexistence through relative non-linearity in their functional responses strongly relies on the Rosenzweig-MacArthur equations (1963) in which the (autotrophic) prey exhibits logistic growth in the absence of the predators. This implies that the prey is limited by a resource which availability is independent of the predators. This assumption does not hold under nutrient limitation where both prey and predators bind resources such as nitrogen or phosphorus in their biomass. Furthermore, the prey’s resource uptake-rate is assumed to be linear and the predator-prey system is considered to be closed. All these assumptions are unrealistic for many natural systems. Here, we show that predator coexistence on a single prey is strongly hampered when the prey and predators indirectly compete for the limiting resource in a flow-through system. In contrast, a non-linear resource uptake rate of the prey slightly promotes predator coexistence. Our study highlights that predator coexistence does not only depend on differences in the curvature of their functional responses but also on the type of resource constraining the growth of their prey. This has far-reaching consequences for the relative importance of fluctuation-dependent and -independent mechanisms of species coexistence in natural systems where autotrophs experience light or nutrient limitation.

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