Lotka-Volterra Type Model Research Articles

Lotka-Volterra models for the market penetration of renewable energy

We studied trend models for the energy and electricity markets and compared their complexity. Was it associated to the ease of policy interventions to control these markets? Using open data, we applied Lotka-Volterra type models to describe the evolution of the market shares of renewable energy and of electricity from renewable sources and observed that for the latter data more complex models were needed. We then searched for characteristics and anomalies in these models that might indicate immunity to and susceptibility for policy interventions towards reaching carbon neutrality, respectively. However, the evidence linking complexity and receptiveness for policy interventions remained weak.

Numerical response of predator to prey: Dynamic interactions and population cycles in Eurasian lynx and roe deer

AbstractThe dynamic interactions between predators and their prey have two fundamental processes: numerical and functional responses. Numerical response is defined as predator growth rate as a function of prey density or both prey and predator densities [dP/dt = f(N, P)]. Functional response is defined as the kill rate by an individual predator being a function of prey density or prey and predator densities combined. Although there are relatively many studies on the functional response in mammalian predators, the numerical response remains poorly documented. We studied the numerical response of Eurasian lynx (Lynx lynx) to various densities of its primary prey species, roe deer (Capreolus capreolus), and to itself (lynx). We exploited an unusual natural situation, spanning three decades where lynx, after a period of absence in central and southern Sweden, during which roe deer populations had grown to high densities, subsequently recolonized region after region, from north to south. We divided the study area into seven regions, with increasing productivity from north to south. We found strong effects of both roe deer density and lynx density on lynx numerical response. Thus, both resources and intraspecific competition for these resources are important to understanding the lynx population dynamic. We built a series of deterministic lynx–roe deer models, and applied them to the seven regions. We found a very good fit between these Lotka–Volterra type models and the data. The deterministic models produced almost cyclic dynamics or dampened cycles in five of the seven regions. Thus, we documented population cycles in this large predator–large herbivore system, which is rarely done. The amplitudes in the dampened cycles decreased toward the south. Thus, the dynamics between lynx and roe deer became more stable with increasing carrying capacity for roe deer, which is related to higher productivity in the environment. This increased stability could be explained by variation in predation risk, where human presence can act as prey refugia, and by a more diverse prey guild that will weaken the direct interaction between lynx and roe deer.

THE ROLE OF HABITAT COUPLING BY ZOOPLANKTON DRIVING POPULATION DYNAMICS AND STABILITY IN SHALLOW LAKES

Studies of population stability in shallow lakes are yet to explain how fishless ponds, with high algae productivity, can have stable zooplankton-algae populations throughout the year. These studies have traditionally overlooked the role of benthic-pelagic coupling, a phenomenon that has noticeable effects on population stability in aquatic environments. We analyzed a simple model to show that benthic-pelagic habitat coupling can explain discrepancies between the behavior of classical predator-prey models and the patterns observed in natural aquatic systems. We used a Lotka-Volterra type model of zooplankton and algae, explicitly modeled as phytoplankton and periphyton. Zooplankton could eat in both algal compartments, presenting a multi-chain omnivore configuration, whereas phytoplankton and periphyton engage in exploitative competition as system support capacity increases. We also modeled the algal exchange among compartments. Our model results show that (1) zooplankton—algae systems tend to be stable up to high nutrient values at intermediate degrees of omnivory, that (2) algae exchange among compartments may dampen stability and that (3) exploratory competition between phytoplankton and the periphyton can also decrease stability. The model results are supported by empirical results available in the literature. Despite the limitations of the modeling approach, our results emphasize the role of habitat coupling and contribute to deepening the understanding of the processes and mechanisms capable of promoting the stability of population dynamics in shallow lakes

Dynamics of a Predator–Prey Model with the Additive Predation in Prey

In this paper, we consider a predator–prey model, in which the prey’s growth is affected by the additive predation of its potential predators. Due to the additive predation term in prey, the model may exhibit the cases of the strong Allee effect, weak Allee effect and no Allee effect. In each case, the dynamics of global features of the model are investigated. Compared to the well-known Lotka–Volterra type model, the model proposed in this paper exhibits much richer and more complex dynamic behaviors, such as the Allee effect, the sensitivity to the initial conditions caused by the strong Allee effect, the oscillatory behavior and the Hopf and heteroclinic bifurcations. Furthermore, the stability and Hopf bifurcation of the model with the density dependent feedback time delay in prey are investigated. By the normal form method and center manifold theory, the explicit formulas are presented to determine the direction of Hopf bifurcation and the stability and period of Hopf-bifurcating periodic solutions. Theoretical analysis and numerical simulation indicate that the delay may destabilize the model, and cause the Hopf bifurcation not only at the interior equilibrium but also at a boundary equilibrium.

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This paper details the approximate solution to the Lotka-Volterra type models with no closed-form by employing the use of two numerical techniques; Runge-Kutta Fehlberg and the 4-stage Runge-Kutta method. From the numerical techniques used, a comparative analysis is carried out, and an approximate solution is obtained. Two cases were considered, case 1 details on indistinguishable graphs but different numerical values (not enough to conclude on the efficient technique) are obtained using the two techniques and while case 2 shows that both numerical techniques give same graphical representation and numerical values, hence this necessitate the reason for this investigation; by so doing the log-plots and residuals was introduced to obtain the most effective and efficient technique under such condition.

Comparative analysis of Lotka-Volterra type models with numerical methods using residuals in Mathematica

Comparative analysis of Lotka-Volterra type models with numerical methods using residuals in Mathematica

A note on the stability of a modified Lotka-Volterra model using Hurwitz polynomials

In the analysis of the dynamics of the solutions of ordinary differential equations we can observe whether or not small variations or perturbations in the initial conditions produce small changes in the future; this intuitive idea of stability was formalized and studied by Lyapunov, who presented methods for the stable analysis of differential equations. For linear or nonlinear systems, we can also analyze the stability using criteria to obtain Hurwitz type polynomials, which provide conditions for the analysis of the dynamics of the system, studying the location of the roots of the associated characteristic polynomial. In this paper we present a stability study of a Lotka-Volterra type model which has been modified considering the carrying capacity or support in the prey and time delay in the predator, this stable analysis is performed using stability criteria to obtain Hurwitz-type polynomials.

Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent

<p style='text-indent:20px;'>We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish <inline-formula><tex-math id="M1">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> time-decay for the linear homogeneous system by using a Fourier transform and finding Green's matrix. Then, we find <inline-formula><tex-math id="M2">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.</p>

Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior.

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.

An Analysis of the Dynamics of a Cancerous Tumour Model with Targeted Chemotherapy

We have analyzed a model of Lotka-Volterra type interacting between immune cell-tumour cell-normal cells, where control policy is applied in terms of targeted chemotherapy. We determined conditions for the local stability of all the equilibrium points and global stability condition for the tumour free equilibrium point, including the feasibility of the solution. Further, we have discussed the possibility of Hopf bifurcation at each equilibrium point. Numerical simulation was carried out to observe the qualitative behaviour of the system as the control parameter is varied.

Chaotic attractors in Atkinson–Allen model of four competing species

We study the occurrence of chaos in the Atkinson–Allen model of four competing species, which plays the role as a discrete-time Lotka–Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark–Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson–Allen dynamics can lead to chaos.

Dynamics of a stoichiometric producer-grazer system with seasonal effects on light level.

Many population systems are subject to seasonally varying environments. As a result, many species exhibit seasonal changes in their life-history parameters. It is quite natural to try to understand how seasonal forcing affects population dynamics subject to stoichiometric constraints, such as nutrient/light availability and food quality. Here, we use a variation of a stoichiometric Lotka-Volterra type model, known as the LKE model, as a case study, focusing on seasonal variation in the producer's light-dependent carrying capacity. Positivity and boundedness of model solutions are studied, as well as numerical explorations and bifurcations analyses. In the absence of seasonal effects, the LKE model suggests that the dynamics are either stable equilibrium or limit cycles. However, through bifurcation analysis we observe that seasonal forcing can lead to complicated population dynamics, including periodic and quasi-periodic solutions.

On existence of wavefront solutions in mixed monotone reaction-diffusion systems

In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.

Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect

The dynamics of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and Dirichlet boundary conditions is investigated in this paper. The existence and multiplicity of spatially nonhomogeneous steady-state solutions are obtained by means of Lyapunov–Schmidt reduction. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.

A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on consumer dynamics

There has been important progress in understanding ecological dynamics through the development of the theory of ecological stoichiometry. For example, modeling under this framework allows food quality to affect consumer dynamics. While the effects of nutrient deficiency on consumer growth are well understood, recent discoveries in ecological stoichiometry suggest that consumer dynamics are not only affected by insufficient food nutrient content (low phosphorus (P): carbon (C) ratio) but also by excess food nutrient content (high P:C). This phenomenon is known as the stoichiometric knife edge, in which animal growth is reduced not only by food with low P content but also by food with high P content, and needs to be incorporated into mathematical models. Here we present a Lotka–Volterra type model to investigate the growth response of Daphnia to algae of varying P:C ratios capturing the mechanism of the stoichiometric knife edge.

A Commensal Consumer-Induced Mediation Effects on Resource–Consumer Interactions

A general view of resource–consumer interactions is that resource intake by the consumers reduces the growth rate of resource population but it leads to an increase of consumer population. This view is proficiently interpreted with the classic Lotka–Volterra model that successfully describes the effects of changes in consumption rates due to changes in resource and consumer population densities. These effects are resulted in perpetual oscillatory dynamics of both the population densities, and the extent of the effects for given initial densities is measured by the oscillating frequency determined by the model parameters. But in many ecosystems, it has often observed a steep decline and delayed recovery in resource population that cannot be explained by the traditional Lotka–Volterra model. Foraging habits and behaviors of a consumer population may facilitate others, those usually do not affect them directly, to feed on the same resource and then to reproduce successfully. Such commensal consumers (facilitated population) can heavily influence the rate of resource exploitation and thereby affect the usual resource–consumer cycles. While involving such commensal consumer-induced effects, called here commensal mediation, into the Lotka–Volterra type models, it shows that the commensal mediation can have stabilizing or destabilizing effects on resource dynamics depending on the strength of interactions and the conditions in which the interactions occur. In the natural ecosystems where the growth rate of resource population depends on its own density even in absence of consumers, the commensal mediation provides a destabilizing effect on resource dynamics; increasing commensal population density increases the amplitude of resource fluctuations and the time laps from one peak to the next. On the other hand, in the managed ecosystems where the growth rate of resource population is expected to be maintained at a constant level in absence of consumers, the commensal mediation provides stabilizing effect at a certain condition; with a given restriction on the consumer population, decreasing mortality of the commensal population can stabilize the resource population dynamics at a stable, steady-state. Moreover, while the resource population experiences saturation effect, resource–consumer interactions with the commensal mediation exhibit a range of dynamical behaviours starting from stable equilibrium, then damped oscillation, to limit cycles as the resource carrying capacity increases from a critical level. In addition, commensal mediations with both controlling facilitator consumer population and resource harvesting are analyzed separately and the results are discussed for some exemplified managed and natural ecosystems.

Global Dynamics of Discrete Competitive Models with Large Intrinsic Growth Rates

The global dynamics of discrete competitive model of Lotka-Volterra type with two species is considered. Earlier works have shown that the unique positive equilibrium is globally attractive under the assumption that the intrinsic growth rates of the two competitive species are less than 1+ln 2, and further the unique positive equilibrium is globally asymptotically stable under the stronger condition that the intrinsic growth rates of the two competitive species are less than or equal to 1. We prove that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1 + ln 2.

Canard doublet in a Lotka-Volterra type model

New types of canard solutions that arise in the modified Lotka-Volterra equations are discussed.

Bistable wavefronts in a diffusive and competitive Lotka–Volterra type system with nonlocal delays

A diffusive Lotka–Volterra type model with nonlocal delays for two competitive species is considered. The existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four-dimensional system without delay. Furthermore, in order to prove the asymptotic stability (up to translation) of bistable wavefronts of the system, the existence, regularity and comparison theorem of solutions of the corresponding Cauchy problem are first established for the systems on R by appealing to the theory of abstract functional differential equations. The asymptotic stability (up to translation) of bistable wavefronts are then proved by spectral methods. In particular, we also prove that the spreading speed is unique by upper and lower solutions technique. From the point of view of ecology, our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation.

Individual-based model for coevolving competing populations

Classical models for competition between two species usually predict exclusion or divergent evolution of resource exploitation. However, recent experimental data show that coexistence is possible for very similar species competing for the same resources without niche partition. Motivated by this experimental challenge to classical competition theory, we propose an individual-based stochastic competition model, which is essentially a modification of a deterministic Lotka–Volterra type model. The proposed model of competition dynamics incorporates the effects of a discrete genotype, which determines the individual's adaptation to the environment, as well as its interaction with the other species.