scholarly journals Controller design techniques for the Lotka-Volterra nonlinear system

2005 ◽  
Vol 16 (2) ◽  
pp. 124-135 ◽  
Author(s):  
Magno Enrique Mendoza Meza ◽  
Amit Bhaya ◽  
Eugenius Kaszkurewicz
Keyword(s):  
Controller Design ◽  
Nonlinear Dynamical System ◽  
Volterra Model ◽  
Predator Population ◽  
Ecological Context ◽  
Mathematical Ecology ◽  
Nonlinear Dynamical ◽  
Predator Prey ◽  
Predator Prey Models ◽  
Design Techniques

A large class of predator-prey models can be written as a nonlinear dynamical system in one or two variables (species). In many contexts, it is necessary to introduce a control into these dynamics. In this paper we focus on models of two species, and assume, as is common in mathematical ecology, that the control corresponds to a proportional removal of the predator population. Six controller design techniques are applied to the Lotka-Volterra model, which is thus used as a benchmark to evaluate and compare these techniques in an ecological context.

scholarly journals Effects of Behavioral Tactics of Predators on Dynamics of a Predator-Prey System

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Hui Zhang ◽  
Zhihui Ma ◽  
Gongnan Xie ◽  
Lukun Jia
Keyword(s):  
Time Scale ◽  
Stable State ◽  
Volterra Model ◽  
Fast Time ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Interaction ◽  
Game Dynamic

A predator-prey model incorporating individual behavior is presented, where the predator-prey interaction is described by a classical Lotka-Volterra model with self-limiting prey; predators can use the behavioral tactics of rock-paper-scissors to dispute a prey when they meet. The predator behavioral change is described by replicator equations, a game dynamic model at the fast time scale, whereas predator-prey interactions are assumed acting at a relatively slow time scale. Aggregation approach is applied to combine the two time scales into a single one. The analytical results show that predators have an equal probability to adopt three strategies at the stable state of the predator-prey interaction system. The diversification tactics taking by predator population benefits the survival of the predator population itself, more importantly, it also maintains the stability of the predator-prey system. Explicitly, immediate contest behavior of predators can promote density of the predator population and keep the preys at a lower density. However, a large cost of fighting will cause not only the density of predators to be lower but also preys to be higher, which may even lead to extinction of the predator populations.

scholarly journals BACKWARD BIFURCATION IN NEUTROPHIL-PATHOGEN INTERACTION

2019 ◽  
Vol 27 (1) ◽  
pp. 141-156
Author(s):  
CHRISTOPHER M. KRIBS ◽  
OMOMAYOWA OLAWOYIN
Keyword(s):  
Dynamical System ◽  
Borrelia Burgdorferi ◽  
Immune Responses ◽  
Bacterial Growth ◽  
Bacterial Infections ◽  
Nonlinear Dynamical System ◽  
Backward Bifurcation ◽  
Nonlinear Dynamical ◽  
Predator Prey ◽  
Bilinear Terms

Bacterial infections elicit immune responses including neutrophils, whose recruitment is stimulated by the bacteria’s presence but which die after eliminating those bacteria. This dual interaction between bacteria and neutrophil concentrations, more complicated than the simple predator-prey relationship that describes macrophage-bacteria interactions, creates an environment in which neutrophils may only be able to clear sufficiently small infections. This study describes this relationship using a simple nonlinear dynamical system which exhibits bistability behavior known as a backward bifurcation. Bacterial growth is assumed limited by a key nutrient. In contrast to a previous study which held neutrophil and nutrient levels constant and required saturation terms to produce bistability, our model shows that simple bilinear terms support bistability when nutrient and neutrophil densities are allowed to vary in response to bacterial density. An example application involving Borrelia burgdorferi, which feeds on manganese, illustrates why neutrophils’ rapid response is key to their ability to contain bacterial infections.

scholarly journals Ratio-Dependence in Predator-Prey Systems as an Edge and Basic Minimal Model of Predator Interference

2021 ◽  
Vol 9 ◽  
Author(s):  
Yuri V. Tyutyunov ◽  
Lyudmila I. Titova
Keyword(s):  
Volterra Model ◽  
Strong Impact ◽  
Natural Ecosystems ◽  
Functional Responses ◽  
Predator Prey ◽  
Ratio Dependence ◽  
Predator Interference ◽  
Trophic Function ◽  
Ratio Dependent ◽  
Predator Prey Models

The functional response (trophic function or individual ration) quantifies the average amount of prey consumed per unit of time by a single predator. Since the seminal Lotka-Volterra model, it is a key element of the predation theory. Holling has enhanced the theory by classifying prey-dependent functional responses into three types that long remained a generally accepted basis of modeling predator-prey interactions. However, contradictions between the observed dynamics of natural ecosystems and the properties of predator-prey models with Holling-type trophic functions, such as the paradox of enrichment, the paradox of biological control, and the paradoxical enrichment response mediated by trophic cascades, required further improvement of the theory. This led to the idea of the inclusion of predator interference into the trophic function. Various functional responses depending on both prey and predator densities have been suggested and compared in their performance to fit observed data. At the end of the 1980s, Arditi and Ginzburg stimulated a lively debate having a strong impact on predation theory. They proposed the concept of a spectrum of predator-dependent trophic functions, with two opposite edges being the prey-dependent and the ratio-dependent cases, and they suggested revising the theory by using the ratio-dependent edge of the spectrum as a null model of predator interference. Ratio-dependence offers the simplest way of accounting for mutual interference in predator-prey models, resolving the abovementioned contradictions between theory and natural observations. Depending on the practical needs and the availability of observations, the more detailed models can be built on this theoretical basis.

Predator–prey interactions

2007 ◽  
Author(s):  
Michael B. Bonsall ◽  
Michael P. Hassell
Keyword(s):  
Population Size ◽  
Positive Impact ◽  
Parasitic Wasp ◽  
Prey Population ◽  
Volterra Model ◽  
Predator Population ◽  
Continuous Time Model ◽  
Predator Prey ◽  
Population Process ◽  
Prey Interaction

Predation is a widespread population process that has evolved many times within the metazoa. It can affect the distribution, abundance, and dynamics of species in ecosystems. For instance, the distribution of western tussock moth is known to be affected by a parasitic wasp (Maron and Harrison, 1997; Hastings et al., 1998), the abundance of different competitors can be shaped by the presence or absence of predators (e.g. Paine, 1966), and natural enemies (such as many parasitoids) can shape the dynamics of a number of ecological interactions (Hassell, 1978, 2000). The broad aim of this chapter is to explore the dynamical effects of predators (including the large groupings of insect parasitoids) and show how our understanding of predator–prey interactions scales from knowledge of the behaviour and local patch dynamics to the population and regional (metapopulation) levels. We draw on a number of approaches including behavioural studies, population dynamics, and time-series analysis, and use models to describe the data and dynamics of the interaction between predators and prey. Predator–prey interactions have an inherent tendency to fluctuate and show oscillatory behaviour. If predators are initially rare, then the size of the prey population can increase. As prey population size increases, the predator populations also begins to increase, which in turn has a detrimental effect on the prey population leading to a decline in prey numbers. As prey become scarce then the predator population size declines and the cycle starts again. These intuitive dynamics can be captured by one of the simplest mathematical descriptions of a predator–prey interaction: the Lotka–Volterra model (Lotka, 1925; Volterra, 1926). Specifically, the Lotka–Volterra model for an interaction between a predator (P) and its prey (N) is a continuous-time model and has the form : where r is the prey-population growth rate in the absence of predators, α is the predator attack rate, c is the (positive) impact of prey on predators, and d is the death rate of predators in the absence of their prey resource.

scholarly journals Qualitative Analysis of a Resource Management Model and Its Application to the Past and Future of Endangered Whale Populations

2021 ◽  
Vol 14 (1) ◽  
pp. 1-18
Author(s):  
Ledder Ledder
Keyword(s):  
Rapid Decline ◽  
Predator Population ◽  
Population Declines ◽  
Phase Line ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Intermediate Consumption ◽  
Stable Equilibria ◽  
Predator Prey Models ◽  
Type 3

Observed whale dynamics show drastic historical population declines, some of which have not been reversed in spite of restrictions on harvesting. This phenomenon is not explained by traditional predator prey models, but we can do better by using models that incorporate more sophisticated assumptions about consumer-resource interaction. To that end, we derive the Holling type 3 consumption rate model and use it in a one-variable differential equation obtained by treating the predator population in a predator-prey model as a parameter rather than a dynamic variable. The resulting model produces dynamics in which low and high consumption levels lead to single high and low-level stable resource equilibria, respectively, while intermediate consumption levels result in both high and low stable equilibria. The phase line analysis is made more transparent by applying a particular structure to the function that gives the derivative in terms of the state. By positing a consumption level that starts low, gradually increases through technological change and human population growth, and decreases as a result of public policy, we are able to tell a story that explains the unexpectedly rapid decline of some resources, such as whales, followed by limited recovery in response to conservation. The analysis also offers guidelines for how to establish sustainable harvesting for restored populations. We include a bifurcation analysis and suggestions for how to teach the material with three different levels of focus on the modeling aspect of the study.

scholarly journals Bifurcation analysis of the predator–prey model with the Allee effect in the predator

2021 ◽  
Vol 84 (1-2) ◽  
Author(s):  
Deeptajyoti Sen ◽  
Saktipada Ghorai ◽  
Malay Banerjee ◽  
Andrew Morozov
Keyword(s):  
Allee Effect ◽  
Mathematical Formulation ◽  
Global Bifurcation ◽  
Predator Population ◽  
Stable Limit ◽  
Bifurcation Structure ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models

AbstractThe use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

scholarly journals Predator-Prey in Tumor-Immune Interactions: A Wrong Model or Just an Incomplete One?

2021 ◽  
Vol 12 ◽  
Author(s):  
Irina Kareva ◽  
Kimberly A. Luddy ◽  
Cliona O’Farrelly ◽  
Robert A. Gatenby ◽  
Joel S. Brown
Keyword(s):  
Immune System ◽  
Immune Cell ◽  
Interference Competition ◽  
Prey Type ◽  
Cell Types ◽  
Cell Populations ◽  
Natural Food ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Models

Tumor-immune interactions are often framed as predator-prey. This imperfect analogy describes how immune cells (the predators) hunt and kill immunogenic tumor cells (the prey). It allows for evaluation of tumor cell populations that change over time during immunoediting and it also considers how the immune system changes in response to these alterations. However, two aspects of predator-prey type models are not typically observed in immuno-oncology. The first concerns the conversion of prey killed into predator biomass. In standard predator-prey models, the predator relies on the prey for nutrients, while in the tumor microenvironment the predator and prey compete for resources (e.g. glucose). The second concerns oscillatory dynamics. Standard predator-prey models can show a perpetual cycling in both prey and predator population sizes, while in oncology we see increases in tumor volume and decreases in infiltrating immune cell populations. Here we discuss the applicability of predator-prey models in the context of cancer immunology and evaluate possible causes for discrepancies. Key processes include “safety in numbers”, resource availability, time delays, interference competition, and immunoediting. Finally, we propose a way forward to reconcile differences between model predictions and empirical observations. The immune system is not just predator-prey. Like natural food webs, the immune-tumor community of cell types forms an immune-web of different and identifiable interactions.

Interspecific Interactions I: Predation and Parasitism

2020 ◽  
pp. 47-66
Author(s):  
Michael J. Fogarty ◽  
Jeremy S. Collie
Keyword(s):  
Aquatic Ecosystems ◽  
Multiple Equilibria ◽  
Interspecific Interactions ◽  
Disease Outbreaks ◽  
Predator Population ◽  
Predator Prey ◽  
Predation Effects ◽  
Human Predation ◽  
Predator Prey Models ◽  
Aquaculture Facilities

Predation and parasitism are dominant forms of interspecific interactions in aquatic ecosystems. Predation effects have been more commonly quantified in aquatic ecosystems than disease. Diet studies documenting predation are substantially more common that routine monitoring for disease in aquaculture systems. The simplest predator–prey models predict lagged cycles of prey and their predators. Density-dependent regulation of the prey or predator population is required for stable coexistence of predator and prey populations. Predator–prey models are extended with the incorporation of non-linear functional responses, which can result in multiple equilibria. The behavior and dynamics of natural predators hold important insights in our consideration of human predation on aquatic resource species. Disease outbreaks have wrought tremendous impacts on a very broad spectrum of aquatic species. For economically important species, these impacts include significant economic costs to fishing communities and aquaculture facilities.

scholarly journals PENYELESAIAN MODIFIKASI MODEL PREDATOR PREY LESLIE-GOWER DENGAN SEBAGIAN PREY TERINFEKSI MENGGUNAKAN ADAMS BASHFORTH MOULTON ORDE EMPAT

2019 ◽  
Vol 19 (2) ◽  
pp. 53
Author(s):  
Liatri Arianti ◽  
Rusli Hidayat ◽  
Kosala Dwija Purnomo
Keyword(s):  
Fourth Order ◽  
Local Stability ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Spreading Model ◽  
Disease Spreading ◽  
Predator Prey Models ◽  
Infected Prey ◽  
Model Predator

Eco-epidemiology is a science that studies the spread of infectious diseases in a population in an ecosystem where two or more species interact like a predator prey. In this paper discusses about how to solve modification Leslie Gower of predator prey models (with Holling II response function) with some prey infected using fourth order Adams Bashforth Moulton method. This paper used a simple disease-spreading model that is Susceptible-Infected (SI). The model is divided into three populations: the sound prey (which is susceptible), the infected prey and predator population. Keywords: Adams Basforth Moulton, Eco-epidemiology Holling Tipe II, Local stability, Leslie-Gower, Predator-Prey model

scholarly journals Stochastic dynamics of predator-prey interactions

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0255880
Author(s):  
Abhyudai Singh
Keyword(s):  
Attack Rate ◽  
Stochastic Dynamics ◽  
Volterra Model ◽  
Reproduction Rate ◽  
Predator Population ◽  
Population Densities ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Predator Attack

The interaction between a consumer (such as, a predator or a parasitoid) and a resource (such as, a prey or a host) forms an integral motif in ecological food webs, and has been modeled since the early 20th century starting from the seminal work of Lotka and Volterra. While the Lotka-Volterra predator-prey model predicts a neutrally stable equilibrium with oscillating population densities, a density-dependent predator attack rate is known to stabilize the equilibrium. Here, we consider a stochastic formulation of the Lotka-Volterra model where the prey’s reproduction rate is a random process, and the predator’s attack rate depends on both the prey and predator population densities. Analysis shows that increasing the sensitivity of the attack rate to the prey density attenuates the magnitude of stochastic fluctuations in the population densities. In contrast, these fluctuations vary non-monotonically with the sensitivity of the attack rate to the predator density with an optimal level of sensitivity minimizing the magnitude of fluctuations. Interestingly, our systematic study of the predator-prey correlations reveals distinct signatures depending on the form of the density-dependent attack rate. In summary, stochastic dynamics of nonlinear Lotka-Volterra models can be harnessed to infer density-dependent mechanisms regulating predator-prey interactions. Moreover, these mechanisms can have contrasting consequences on population density fluctuations, with predator-dependent attack rates amplifying stochasticity, while prey-dependent attack rates countering to buffer fluctuations.

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