scholarly journals Evolutionary distributions in adaptive space

2005 ◽  
Vol 2005 (4) ◽  
pp. 403-424 ◽  
Author(s):  
Yosef Cohen
Keyword(s):  
Complete Separation ◽  
The Other ◽  
Adaptive Traits ◽  
Predator Prey ◽  
Prey System ◽  
Algebraic Problem ◽  
Modeling Systems ◽  
Evolutionary Context ◽  
Dynamics Of Population ◽  
Predator Prey Models

An evolutionary distribution (ED), denoted byz(x,t), is a distribution of density of phenotypes over a set of adaptive traitsx. Herexis ann-dimensional vector that represents the adaptive space. Evolutionary interactions among phenotypes occur within an ED and between EDs. A generic approach to modeling systems of ED is developed. With it, two cases are analyzed. (1) A predator prey inter-ED interactions either with no intra-ED interactions or with cannibalism and competition (both intra-ED interactions). A predator prey system with no intra-ED interactions is stable. Cannibalism destabilizes it and competition strengthens its stability. (2) Mixed interactions (where phenotypes of one ED both benefit and are harmed by phenotypes of another ED) produce complete separation of phenotypes on one ED from the other along the adaptive trait. Foundational definitions of ED, adaptive space, and so on are also given. We argue that in evolutionary context, predator prey models with predator saturation make less sense than in ecological models. Also, with ED, the dynamics of population genetics may be reduced to an algebraic problem. Finally, extensions to the theory are proposed.

scholarly journals Boundedness and Global Stability for a Predator-Prey System with the Beddington-DeAngelis Functional Response

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
Wahiba Khellaf ◽  
Nasreddine Hamri
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Qualitative Behavior ◽  
Uniform Persistence ◽  
Boundedness Of Solutions ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Prey System ◽  
Predator Prey Models ◽  
Type Ii Functional Response

We study the qualitative behavior of a class of predator-prey models with Beddington-DeAngelis-type functional response, primarily from the viewpoint of permanence (uniform persistence). The Beddington-DeAngelis functional response is similar to the Holling type-II functional response but contains a term describing mutual interference by predators. We establish criteria under which we have boundedness of solutions, existence of an attracting set, and global stability of the coexisting interior equilibrium via Lyapunov function.

scholarly journals Optimal dynamic control of predator–prey models

2019 ◽  
Vol 28 (2) ◽  
pp. 425-440 ◽  
Author(s):  
Otwin Becker ◽  
Ulrike Leopold-Wildburger
Keyword(s):  
Dynamic Control ◽  
Decision Support Tool ◽  
Analytical Models ◽  
Support Tool ◽  
Predator Prey ◽  
Sustainable Environment ◽  
Dynamics Of Population ◽  
Optimal Population ◽  
Optimal Dynamic ◽  
Predator Prey Models

Abstract This paper combines work to use a decision support tool for sustainable economic development, while acknowledging interdependent dynamics of population density, and interferences from outside. We get new insights derived from experimental approaches: analytical models (optimal dynamic control of predator–prey models) provide optimal dynamic strategies and interventions, depending on different objective functions. Our economic experiments are able to test the applicability of these strategies, and in how far decision-makers can learn to improve decision-making by repeated applications. We aim to analyse a sustainable environment with diametrical goals to harvest as much as possible while allowing optimal population growth. We find interesting insights from those who manage the dynamic system. With the methodology of experimental economics, the experiment at hand is developed to analyse the capability of individual persons to handle a complex system, and to find an economic, stable equilibrium in a neutral setting. We have developed a most interesting simulation model, where it will turn out that prices play a less important role than availability of the goods. This aspect could become a new important aspect in economics in general and in sustainable environments especially.

SPATIAL PATTERN IN A PREDATOR-PREY SYSTEM WITH BOTH SELF- AND CROSS-DIFFUSION

2009 ◽  
Vol 20 (01) ◽  
pp. 71-84 ◽  
Author(s):  
GUI-QUAN SUN ◽  
ZHEN JIN ◽  
YI-GUO ZHAO ◽  
QUAN-XING LIU ◽  
LI LI
Keyword(s):  
Spatial Patterns ◽  
Temporal Dynamics ◽  
Spatial Scales ◽  
Natural Populations ◽  
Spatial Dynamics ◽  
Density Variation ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Dynamics Of Population

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

scholarly journals Dynamics in two competing predators-one prey system with two types of Holling and fear effect

2021 ◽  
Vol 2 (2) ◽  
pp. 58-67
Author(s):  
Adin Lazuardy Firdiansyah ◽  
Nurhidayati Nurhidayati
Keyword(s):  
Consumption Rate ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
The Other ◽  
Type I ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Local Stability Analysis ◽  
Prey System

In this article, it is formulated a predator-prey model of two predators consuming a single limited prey resource. On the other hand, two predators have to compete with each other for survival. The predation function for two predators is assumed to be different where one predator uses Holling type I while the other uses Holling type II. It is also assumed that the fear effect is considered in this model as indirect influence evoked by both predators. Non-negativity and boundedness is written to show the biological justification of the model. Here, it is found that the model has five equilibrium points existed under certain condition. We also perform the local stability analysis on the equilibrium points with three equilibrium points are stable under certain conditions and two equilibrium points are unstable. Hopf bifurcation is obtained by choosing the consumption rate of the second predator as the bifurcation parameter. In the last part, several numerical solutions are given to support the analysis results.

Qualitative analysis of a ratio-dependent predator–prey system with diffusion

2003 ◽  
Vol 133 (4) ◽  
pp. 919-942 ◽  
Author(s):  
Peter Y. H. Pang ◽  
Mingxin Wang
Keyword(s):  
Differential Equation ◽  
Differential System ◽  
Neumann Boundary ◽  
Qualitative Properties ◽  
Partial Differential ◽  
Predator Prey ◽  
Prey System ◽  
Homogeneous Neumann Boundary Condition ◽  
Ratio Dependent ◽  
Predator Prey Models

Ratio-dependent predator–prey models are favoured by many animal ecologists recently as they better describe predator–prey interactions where predation involves a searching process. When densities of prey and predator are spatially homogeneous, the so-called Michaelis–Menten ratio-dependent predator–prey system, which is an ordinary differential system, has been studied by many authors. The present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous Neumann boundary condition. Its main purpose is to study qualitative properties of solutions to this reaction-diffusion (partial differential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-differential-equation dynamics, non-constant positive steady states exist for the partial-differential-equation model. This demonstrates that stationary patterns arise as a result of diffusion.

scholarly journals Existence of Positive Periodic Solutions for a Predator-Prey System of Holling Type IV Function Response with Mutual Interference and Impulsive Effects

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Haidong Liu ◽  
Fanwei Meng
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Mutual Interference ◽  
Impulsive Effects ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Type Iv ◽  
Prey System ◽  
Predator Prey Models

We investigate the existence of periodic solutions for a predator-prey system with Holling function response and mutual interference. Our model is more general than others since it has both Holling type IV function and impulsive effects. With some new analytical tricks and the continuation theorem in coincidence degree theory proposed by Gaines and Mawhin, we obtain a set of sufficient conditions on the existence of positive periodic solutions for such a system. In addition, in the remark, we point out some minor errors which appeared in the proof of theorems in some published papers with relevant predator-prey models. An example is given to illustrate our results.

On a delay ratio-dependent predator–prey system with feedback controls and shelter for the prey

2018 ◽  
Vol 11 (07) ◽  
pp. 1850095
Author(s):  
Changyou Wang ◽  
Linrui Li ◽  
Yuqian Zhou ◽  
Rui Li
Keyword(s):  
Numerical Solutions ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Feedback Controls ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Ratio Dependent ◽  
Predator Prey Models

In this paper, a class of three-species multi-delay Lotka–Volterra ratio-dependent predator–prey model with feedback controls and shelter for the prey is considered. A set of easily verifiable sufficient conditions which guarantees the permanence of the system and the global attractivity of positive solution for the predator–prey system are established by developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function. Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In addition, some numerical solutions of the equations describing the system are given to show that the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator–prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system. At the same time, the influence of the delays and shelters on the dynamics behavior of the system is also considered by solving numerically the predator–prey models.

scholarly journals Deterministic and Stochastic Analysis of a Ratio-Dependent Predator-Prey System with Delay

2007 ◽  
Vol 12 (3) ◽  
pp. 383-398 ◽  
Author(s):  
A. Maiti ◽  
M. M. Jana ◽  
G. P. Samanta
Keyword(s):  
Time Delay ◽  
Computer Simulations ◽  
Discrete Time ◽  
Stochastic Analysis ◽  
Predator Prey ◽  
Theoretical Ecology ◽  
Prey System ◽  
Discrete Time Delay ◽  
Ratio Dependent ◽  
Predator Prey Models

Recently ratio-dependent predator-prey models have become the focus of considerable attention in theoretical ecology in their own right. In this paper, we have studied the deterministic and stochastic dynamical aspects of stability of a MichaelisMenten type ratio-dependent predator-prey system that includes discrete time-delay. Computer simulations are carried out to explain the analytical findings in deterministic environment. The biological implications of our analytical and numerical findings are discussed critically.

scholarly journals A study of a three-dimensional competitive Lotka–Volterra system

2020 ◽  
Vol 34 ◽  
pp. 03010
Author(s):  
Florian Munteanu
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Local Behavior ◽  
Real Numbers ◽  
Volterra System ◽  
Positive Real ◽  
Predator Prey ◽  
Strictly Positive Real ◽  
Dynamics Of Population ◽  
Predator Prey Models

In this paper we will consider a community of three mutually competing species modeled by the Lotka–Volterra system: $$ {\left\{ {\dot x} \right._i} = {x_i}\left( {{b_i} - \sum\limits_{i = 1}^3 {{a_{ij}}{x_j}} } \right),i = 1,2,3 $$ where xi(t) is the population size of the i-th species at time t, Ẋi denote $${{dxi} \over {dt}}$$ and aij, bi are all strictly positive real numbers. This system of ordinary differential equations represent a class of Kolmogorov systems. This kind of systems is widely used in the mathematical models for the dynamics of population, like predator-prey models or different models for the spread of diseases. A qualitative analysis of this Lotka-Volterra system based on dynamical systems theory will be performed, by studying the local behavior in equilibrium points and obtaining local dynamics properties.

A GENERALIZED PREDATOR–PREY SYSTEM WITH HABITAT COMPLEXITY

2017 ◽  
Vol 25 (03) ◽  
pp. 495-520 ◽  
Author(s):  
ZHIHUI MA ◽  
SHUFAN WANG ◽  
TINGTING WANG ◽  
HAOPENG TANG ◽  
ZIZHEN LI
Keyword(s):  
Numerical Simulation ◽  
Habitat Complexity ◽  
The Other ◽  
Predator Prey ◽  
Prey System ◽  
Other Hand ◽  
Stabilizing Effect ◽  
Open Issue ◽  
The One

This paper presents a generalized predator–prey system and considers the effect of habitat complexity on the dynamical consequences. The results show that habitat complexity has a major impact on the dynamical consequences of the considered system. On the one hand, habitat complexity has a stabilizing impact under certain conditions. A numerical simulation in our study and in experiments conducted in the published studies elaborate on this stabilizing effect. On the other hand, the most interesting and open issue is that a destabilizing effect of habitat complexity is found theoretically. All results are explained and illustrated from the ecological viewpoint.

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