scholarly journals COMPLEX BEHAVIOR IN SIMPLE MODELS OF BIOLOGICAL COEVOLUTION

2009 ◽  
Vol 20 (09) ◽  
pp. 1387-1397 ◽  
Author(s):  
PER ARNE RIKVOLD
Keyword(s):  
Kinetic Monte Carlo ◽  
Dynamical Behavior ◽  
Individual Species ◽  
Predator Prey ◽  
Population Sizes ◽  
Kinetic Monte Carlo Simulations ◽  
Predator Prey Models ◽  
Power Law Distributions ◽  
Adaptive Foraging ◽  
Type Ii Functional Response

We explore the complex dynamical behavior of simple predator-prey models of biological coevolution that account for interspecific and intraspecific competition for resources, as well as adaptive foraging behavior. In long kinetic Monte Carlo simulations of these models we find quite robust 1/f-like noise in species diversity and population sizes, as well as power-law distributions for the lifetimes of individual species and the durations of quiet periods of relative evolutionary stasis. In one model, based on the Holling Type II functional response, adaptive foraging produces a metastable low-diversity phase and a stable high-diversity phase.

Extinction probabilities in predator–prey models

1986 ◽  
Vol 23 (01) ◽  
pp. 1-13
Author(s):  
S. E. Hitchcock
Keyword(s):  
Stochastic Models ◽  
Birth Rate ◽  
Exact Expression ◽  
Initial Population ◽  
Predator Prey ◽  
Extinction Probabilities ◽  
Population Sizes ◽  
The Mean ◽  
Predator Prey Models ◽  
Two Populations

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

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.

Extinction probabilities in predator–prey models

1986 ◽  
Vol 23 (1) ◽  
pp. 1-13 ◽  
Author(s):  
S. E. Hitchcock
Keyword(s):  
Stochastic Models ◽  
Birth Rate ◽  
Exact Expression ◽  
Initial Population ◽  
Predator Prey ◽  
Extinction Probabilities ◽  
Population Sizes ◽  
The Mean ◽  
Predator Prey Models ◽  
Two Populations

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

Model for the catalytic reduction of no on a surface with species desorption and impurities that cannot desorb

2021 ◽  
Author(s):  
E. J. Hernández ◽  
G. M. Buendía
Keyword(s):  
Catalytic Reduction ◽  
Kinetic Monte Carlo ◽  
Dynamical Behavior ◽  
Square Lattice ◽  
Modified Model ◽  
Reduction Of No ◽  
Large Fluctuations ◽  
Long Time ◽  
Kinetic Monte Carlo Simulations ◽  
The Stability

The dynamical behavior of a modified Yaldram–Khan model for the catalytic reduction of NO on a surface is studied by Kinetic Monte Carlo simulations. In this modified model, temperature effects are incorporated as desorption rates of the N and CO species. How the presence of contaminants in the gas phase affects the catalytic process is also analyzed by including impurities that can be adsorbed on the lattice and once there remain inert. When N desorption is included, a reactive window appears that is not present in the original YK model on a square lattice. When CO desorption is added large fluctuations appear in the coverages, the system can take a long time to stabilize, during this period, a long lasting reactive state exists that disappears when the stability is reached. When nondesorbing impurities are added, the discontinuous transition to a CO poisoned phase that presents the original YK model disappears, the coverages become continuous, and a nonreactive steady-state is rapidly reached.

scholarly journals Complex Dynamical Behavior of Holling–Tanner Predator-Prey Model with Cross-Diffusion

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-14
Author(s):  
Caiyun Wang ◽  
Yongyong Pei ◽  
Yaqun Niu ◽  
Ruiqiang He
Keyword(s):  
Dynamical Behavior ◽  
Multiple Scale ◽  
Amplitude Equations ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Pattern Structures ◽  
Exact Distributions ◽  
Predator Prey Models

Spatial predator-prey models have been studied by researchers for many years, because the exact distributions of the population can be well illustrated via pattern formation. In this paper, amplitude equations of a spatial Holling–Tanner predator-prey model are studied via multiple scale analysis. First, by amplitude equations, we obtain the corresponding intervals in which different kinds of patterns will be onset. Additionally, we get the conclusion that pattern transitions of the predator are induced by the increasing rate of conversion into predator biomass. Specifically, pattern transitions of the predator between distinct Turing pattern structures vary in an orderly manner: from spotted patterns to stripe patterns, and finally to black-eye patterns. Moreover, it is discovered that pattern transitions of prey can be induced by cross-diffusion; that is, patterns of prey transmit from spotted patterns to stripe patterns and finally to a mixture of spot and stripe patterns. Meanwhile, it is found that both effects of cross-diffusion and interaction between the prey and predator can lead to the complicated phenomenon of dynamics in the system of biology.

Pattern dynamics of a diffusive predator–prey model with delay effect

2017 ◽  
Vol 10 (04) ◽  
pp. 1750059 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Dongliang Li
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Wave Pattern ◽  
Turing Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Predator Prey Models ◽  
Parameter Values

We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing–Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing–Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction–diffusion system might bring some insight to the finding of patterns in spatial predator–prey models.

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