Three-species Lotka-Volterra Model Research Articles

Chaos emergence and dissipation in a three-species food web model with intraguild predation and cooperative hunting

<abstract><p>We explore the dynamics of a three-species Lotka-Volterra model incorporating intraguild (IG) predation. The model encompasses interactions between a basal prey, intraguild prey and omnivorous top/intraguild predator. These interactions are characterized by linear functional responses, while considering intraspecific competition and cooperative hunting dynamics. The study involves a comprehensive stability of different steady states and bifurcation analysis. Bifurcation structures unveil shifts in equilibrium stability and the emergence of new equilibrium states. Investigation into dynamics around the coexistence equilibrium highlights diverse behaviors, including stable coexistence, oscillations and chaos. Furthermore, exploration of species' densities under parameter variations uncovers distinct patterns, ranging from stability to chaos. Incorporating the concept of hunting cooperation among IG predators and IG prey can lead to the emergence or suppression of chaotic oscillations, respectively. Additionally, we observe that lower consumption rate of IG predator and cooperation of IG predator helps the system to keep in a stable state position.</p></abstract>

Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators

In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.

Period Estimation and Noise in a Neutrally Stable Stochastic Oscillator

The periods of the orbits for the well-mixed cyclic three-species Lotka-Volterra model far away from the fixed point are studied. For finite system sizes, a discrete stochastic approach is employed and periods are found via wavelet analysis. As the system size is increased, a hierarchy of approximations ranging from Poisson noise to Gaussian noise to deterministic models are utilized. Based on the deterministic equations, a mathematical relationship between a conserved quantity of the model and the period of the population oscillations is found. Exploiting this property we then study the deterministic conserved quantity and period noise in finite size systems.

Spatial patterns and biodiversity in off-lattice simulations of a cyclic three-species Lotka-Volterra model

Stochastic simulations of cyclic three-species spatial predator-prey models are usually performed in square lattices with nearest-neighbour interactions starting from random initial conditions. In this letter we describe the results of off-lattice Lotka-Volterra stochastic simulations, showing that the emergence of spiral patterns does occur for sufficiently high values of the (conserved) total density of individuals. We also investigate the dynamics in our simulations, finding an empirical relation characterizing the dependence of the characteristic peak frequency and amplitude on the total density. Finally, we study the impact of the total density on the extinction probability, showing how a low population density may jeopardize biodiversity.

Bifurcation and optimal harvesting of a diffusive predator–prey system with delays and interval biological parameters

This paper deals with a delayed reaction–diffusion three-species Lotka–Volterra model with interval biological parameters and harvesting. Sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Then an optimal control problem has been considered. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical evidence shows that the presence of harvesting can impact the existence of species and over harvesting can result in the extinction of the prey or the predator which is in line with reality.

Facilitation of intraguild prey by its intraguild predator in a three-species Lotka–Volterra model

Explaining the coexistence of multiple species in the competition and predation theatre has proven a great challenge. Traditional intraguild predation (IGP) models have only relatively small regions of stable coexistence of all species. Here, we investigate potential additional mechanisms that extend these regions of stable coexistence. We used a 3-species Lotka–Volterra system to which we added an interaction term to model a unidirectional facilitative relationship between the two predators in the IGP. In this modelling study the IG predator was able to precondition a part of the common resource by an instantaneous manipulation, which resulted in the immobilization of the resource species. This mechanism of immobilization facilitated the resource uptake by the IG prey and thus increased its growth rates even in the presence of the common predator. The facilitative relationship of the IG prey by the IG predator produced a stable coexistence of both predators even though the IG prey was an inferior competitor for a common resource, which cannot be attained with the traditional IGP models. Furthermore, our model predicted a 3-species stable coexistence even at high enrichment where no coexistence was found in the basic IGP model. Thus, we showed that diversity of resource traits could significantly alter emergent community patterns via shifts in exploitative competition of IGP-related predators. The described mechanism could potentially lead to a higher efficiency in exploitation of common resources and thus promote higher diversity in a real community.

Effects of plant availability and habitat size on the coexistence of two competing parasitoids in a tri-trophic food web of canola, diamondback moth and parasitic wasps

Using experiments and spatial modeling, we here show the conditions for two competing parasitoids to coexist under two levels of plant availability in a tri-trophic food web of canola, diamondback moth (Plutella xylostella) and two parasitic wasps (Diadegma semiclausum and Cotesia vestalis) that target the same life stage of the herbivore diamondback moth. Plant availability had a significant effect on both the abundances of parasitoids and the herbivore. However, parasitoid abundances were not different with or without the occurrence of other parasitoid species, suggesting no effect of parasitoid coexistence on their abundances. Using a three-species Lotka–Volterra model, we confirmed the competitive exclusion of species with low consumption rate (=initial population size×per capita consumption rate) in a well-mixed habitat patch. We further built a cellular automaton (CA) to explore the minimum habitat size that ensures the coexistence of these two parasitoids, assuming different searching efficiencies (i.e. different radiuses of searching area and movement abilities for locating the host). With the in-patch dynamics portrayed by the three-species Lotka–Volterra model, the CA model demonstrated the possibility for the two parasitoids to coexist when the number of habitat patches reached a certain number. This research thus highlights the importance of both plant availability and habitat size to the dynamics and stability of the tri-trophic food web.

Traveling waves in nonlocal diffusion systems with delays and partial quasi-monotonicity

This paper is devoted to the study of a three-dimensional delayed system with nonlocal diffusion and partial quasi-monotonicity. By developing a new definition of upper-lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. The results are applied to a nonlocal diffusion model which takes the three-species Lotka-Volterra model as its special case.

Coexistence in a competitor–competitor–mutualist model

This paper is concerned with a system modeling a competitor–competitor–mutualist three-species Lotka–Volterra model. By Schauder fixed point theory, the existence of positive solutions to a strongly coupled elliptic system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed and a numerical simulation is also presented. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak.

Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion

In this paper, we study a diffusive three-species ratio-dependent food chain model, using differential inequality, to obtain sufficient conditions that ensure the permanence of the system and the extinction of predator species. Our results reinforce the main result of Sun Wen, Shihua Chen and Huihai Mei [Positive periodic solution of a more realistic three-species Lotka–Volterra model with delay and density regulation, Chaos, Solitons and Fractals, in press].

Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation

In the paper, a more realistic three-species ratio-dependent Lotka-Volterra model with delay and diffusive and density regulation is investigated. By mean of the powerful and effective coincidence degree theory, we establish sufficient conditions for the existence of at least one positive periodic solution of the model. What’s more, the conditions are easily verifiable.

Coexistence and asymptotic periodicity in a competitor–competitor–mutualist model

In this paper, the competitor–competitor–mutualist three-species Lotka–Volterra model is discussed. Firstly, by Schauder fixed point theory, the coexistence state of the strongly coupled system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak. Secondly, the existence and asymptotic behavior of T-periodic solutions for the periodic reaction–diffusion system under homogeneous Dirichlet boundary conditions are investigated. Sufficient conditions which guarantee the existence of T-periodic solution are also obtained.

OMNIVORY CREATES CHAOS IN SIMPLE FOOD WEB MODELS

Omnivory, defined as feeding on more than one trophic level, was considered rare in nature because of its destabilizing effect. However, recent elaborate studies of natural food webs have shown that omnivory is ubiquitous. It is well known that a simple food chain model of three trophic levels can exhibit chaos if the functional responses are nonlinear. We investigate a three-species Lotka-Volterra model of omnivory in which a predator and a prey share the same resource (i.e., intraguild predation). We demonstrate that intraguild predation sometimes destabilizes food webs and induces chaos, even if the functional responses are linear. The route to chaos is the familiar period-doubling cascade. We compare our results to previous work and discuss the mechanism by which omnivory induces chaos.

Species Packing and Niche Complementarity in Three Sunfishes

A fundamental problem in ecology is to identify and relate those factors which determine the number of species that can stably coexist in an area. Interspecific competition is one such factor, and in principle it limits the similarity of competing species in relation to the abundance and diversity of critical resources. MacArthur and Levins (1967) first explored this limit with a three-species Lotka-Volterra model. The success of a species invading between two residents on a resource axis was related to the differences in mean utilization on this axis. Resource abundance was considered uniform across types, and utilization functions were assumed to be normal, differing only in position on the axis. The analysis of this problem has subsequently been extended to include factors such as environmental variability, the relative abundance of resources, the shape of the utilization curve, and predation pressure (e.g., May and MacArthur 1972; MacArthur 1972a; Roughgarden 1974b; Roughgarden and Feldman 1975). The variety of resources used by a species (i.e., niche width) is a central issue in this theory and is usually treated as a given parameter. Both the niche width and location of a species are related to functional morphology (i.e., the tools used in obtaining resources). In fact, one of the first observations suggesting the direction of the theory was Hutchinson's (1959) remarks on the uniformity of character displacement ratios among a wide variety of sympatric species, particularly in trophic apparatus. Morphological measurements have been widely used since as relative indices of competition or niche width (e.g., Cody 1974a; Van Valen 1965; Soule and Stewart 1970). Very seldom, however, have the differences in foraging efficiency of morphologically similar species been quantified in relation to a set of resources that change in a recognizable (hopefully orderable) manner. Careful studies in this direction would facilitate the testing and development of theory by providing a link between morphology, the efficiency of resource utilization, and the parameters of concern, usually measures of overlap in resource use. In this study I quantitatively relate the body plan in fishes to foraging efficiency on the food size dimension of the niche. Using methods derived from optimal foraging theory, cost curves are constructed on this coordinate for