Two-species Competition Model Research Articles

Random dispersal versus fitness-dependent dispersal

This work extends previous work (Cantrell et al., 2008 [9]) on fitness-dependent dispersal for a single species to a two-species competition model. Both species have the same population dynamics, but one species adopts a combination of random and fitness-dependent dispersal and the other adopts random dispersal. Global existence of smooth solutions to the time-dependent quasilinear parabolic system is studied. When a single species has a strong tendency to move up its fitness gradient, it results in a stable equilibrium that can approximate the spatial distribution predicted by the ideal free distribution (Cantrell et al., 2008 [9]). For the two-species competition model, if one species has strong tendency to move up its fitness gradient, such approximately ideal free dispersal is advantageous relative to random dispersal. Bifurcation analysis shows that two competing species can coexist when one species has only an intermediate tendency to move up its fitness gradient and the other species has a smaller random dispersal rate.

Competition and evolution along environmental gradients: patterns, boundaries and sympatric divergence

The present study examined how competitive interactions and environmental conditions generate species boundaries and determine species distributions. A spatially explicit, quantitative genetic, two-species competition model was used to manipulate the strengths of competition, gene flow and local adaptation along environmental gradients. This allowed us to assess the long-term persistence of each species and whether the ranges they inhabited had boundaries in space or were unlimited. We found that a species boundary arises along less steep environmental gradients when the strength of stabilizing selection and diversifying selection are similar. We also found that a species boundary may arise along shallow environmental gradients if interspecific competition is more intense than intraspecific, which relaxes previous requirements for steep gradients for generating range limits. We determined an analytical form for the critical environmental gradient as a function of ecological and genetic parameters at which a species boundary is expected to arise by competition. Results suggest an alternative to resource competition as an explanation for phenotypic divergence between sympatric competitors. Competitors sharing a trait that is under stabilizing selection along an environmental gradient may segregate spatially and evolve in different regions, with phenotypic sympatric divergence reflecting the resulting clines. Along various types of environmental gradients, variation in stabilizing selection intensities could lead to contrasting patterns in the distribution of species. For stabilizing selection strengths in accord with field data estimates, this study predicts that the level of sympatric character divergence would be limited along environmental gradients.

Pre-images of invariant sets of a discrete-time two-species competition model

In this paper, we explore the structure of pre-images of invariant sets of a discrete-time two-species competition model with singularity at the origin. We first show that this competition model is persistent with respect to the total population of two species, i.e. all initial conditions in are attracted to a compact set which is bounded away from the origin. Then we study the properties of pre-images of this system and give the explicit structure of all pre-images of invariant sets for this system under certain parameter range. These results are analogous to the one-dimensional discrete system. Our study is the first step to explore the structure of the basins of attractions of interior attractors of a general discrete-time two-species model (e.g. the locally asymptotically stable interior period-2 orbit). Finally, we discuss how our results give useful insights on the future study for coexistence of the species and list some open problems related to our system.

Travelling wave solutions of a free boundary problem for a two-species competitive model

We study a di usive logistic system with a free boundary in ecologyproposed by Mimura, Yamada and Yotsutani [10]. Motivated by the spreading-vanishingdichotomy obtained by Du and Lin [1], we suppose the spreadingspeed of the free boundary tends to a constant as time tends to in nity andconsider the corresponding travelling wave problem. We establish the existenceand uniqueness of a travelling wave solution for this free boundary problem.

COMPETING POPULATION MODEL WITH NONLINEAR INTRASPECIFIC REGULATION AND MATURATION DELAYS

This paper considers the combined effects of the nonlinear intra-specific regulation and maturation delays on the two-species competition model. Dynamical behaviors of the model are studied, and sharp global asymptotical stability criteria for the coexistence equilibrium as well as the excluding equilibria are established. It is shown that increase of the maturation delay of one species has negative effect on its permanence and a sufficiently large maturation delay will directly lead to its extinction, and that variation of the intra-specific regulation parameter of one species may change the surviving or extinction behaviors of its competitor.

A unified mechanistic model of niche, neutrality and violation of the competitive exclusion principle

AbstractThe origin of species richness is one of the most widely discussed questions in ecology. The absence of unified mechanistic model of competition makes difficult our deep understanding of this subject. Here we show such a two-species competition model that unifies (i) a mechanistic niche model, (ii) a mechanistic neutral (null) model and (iii) a mechanistic violation of the competitive exclusion principle. Our model is an individual-based cellular automaton. We demonstrate how two trophically identical and aggressively propagating species can stably coexist in one stable homogeneous habitat without any trade-offs in spite of their 10% difference in fitness. Competitive exclusion occurs if the fitness difference is significant (approximately more than 30%). If the species have one and the same fitness they stably coexist and have similar numbers. We conclude that this model shows diffusion-like and half-soliton-like mechanisms of interactions of colliding population waves. The revealed mechanisms eliminate the existing contradictions between ideas of niche, neutrality and cases of violation of the competitive exclusion principle.

Front dynamics in a two-species competition model driven by Lévy flights

A number of recent studies suggest that many biological species follow a Lévy random walk in their search for food. Such a strategy has been shown to be more efficient than classical Brownian motion when resources are scarce. However, current diffusion–reaction models used to describe many ecological systems do not account for the superdiffusive spread of populations due to Lévy flights. We have developed a model to simulate the spatial spread of two species competing for the same resources and driven by Lévy flights. The model is based on the Lotka–Volterra equations and has been obtained by replacing the second-order diffusion operator by a fractional-order one. Consistent with previous known results, theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the population fronts and a power-law decay of the fronts' leading tail. Depending on the skewness of the fractional derivative, we derive catch-up conditions for different types of fronts. Our results indicate that second-order diffusion–reaction models are not well-suited to simulate the spatial spread of biological species that follow a Lévy random walk as they are inclined to underestimate the speed at which these species propagate.

A discrete stage-structured two-species competition model with sexual and clonal reproduction

In this paper, we analyse a discrete stage-structured model which is a generalization of the two-species competition model studied in [2]. Motivated by plant populations, each species is assumed to reproduce both sexually and clonally. We show that this model has a dynamical behaviour that is similar to that of the classical continuous two-dimensional Lotka–Volterra model under weak nonlinearities of the Beverton–Holt type. By allowing the species to have different competition efficiencies, we show that it is possible to obtain different dynamics including coexistence, bistability and competitive exclusion, in contrast with the model studied in [2], which exhibits only competitive exclusion behaviour.

Weak Allee effects and species coexistence

In this article, we study the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects and/or mate limitation induced Allee effects. We focus on the following two possible outcomes of the competition: 1. one species goes to extinction; 2. the system is permanent. Our results indicate that, even if one species’ intra-specific competition is less than its inter-specific competition, weak Allee effects induced by predation saturation can promote coexistence of the two competing species. This is supported by the outcome of two-species competition models without Allee effects. Also, we discuss our results and future work on multiple attractors in competition models with Allee effects.

Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.

Parameter Estimates in Differential Equation Models for Population Growth

We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer Mathematica code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist, G. F. Gause, and estimate growth rates, carrying capacities, and “coefficients for the struggle for existence.”

Trajectory attractors of energy balance climate models with bio-feedback

Motivated by coupling an energy balance climate model and a two-species competition model for the bio-sphere, one is led to the study of functional reaction-diffusion equations with memory and a nonlocal Volterra operator. The existence of a trajectory attractor is established. The work is motivated by similar studies in (12) for a energy balance model with latent heat flux and uses techniques developed in (11) and (12). It is a continuation of (18), where an abstract global existence and boundedness result was established.

Coexistence and invasibility in a two-species competition model with habitat-preference

The outcome of competition among species is influenced by the spatial distribution of species and effects such as demographic stochasticity, immigration fluxes, and the existence of preferred habitats. We introduce an individual-based model describing the competition of two species and incorporating all the above ingredients. We find that the presence of habitat preference—generating spatial niches—strongly stabilizes the coexistence of the two species. Eliminating habitat preference—neutral dynamics—the model generates patterns, such as distribution of population sizes, practically identical to those obtained in the presence of habitat preference, provided an higher immigration rate is considered. Notwithstanding the similarity in the population distribution, we show that invasibility properties depend on habitat preference in a non-trivial way. In particular, the neutral model results more invasible or less invasible depending on whether the comparison is made at equal immigration rate or at equal distribution of population size, respectively. We discuss the relevance of these results for the interpretation of invasibility experiments and the species occupancy of preferred habitats.

On a Nonlocal Reaction-Diffusion Problem Arising from the Modeling of Phytoplankton Growth

In this paper we analyze a nonlocal reaction-diffusion model which arises from the modeling of competition of phytoplankton species with incomplete mixing in a water column. The nonlocal nonlinearity in the model describes the light limitation for the growth of the phytoplankton species. We first consider the single-species case and obtain a complete description of the long-time dynamical behavior of the model. Then we study the two-species competition model and obtain sufficient conditions for the existence of positive steady states and uniform persistence of the dynamical system. Our approach is based on a new modified comparison principle, fixed point index theory, global bifurcation arguments, elliptic and parabolic estimates, and various analytical techniques.

A free boundary problem for two-species competitive model in ecology

This paper deals with a free boundary problem which is used to describe the two-species competitive model in ecology. The existence and uniqueness of a global classical solution are given by invoking the Schauder fixed point theorem. We study the evolution of the free boundary problem and show that the free boundary problem is well posed.

Some remarks on traveling wave solutions in competition models

We study the existence of traveling wave solutions for competition models in the form of integro-difference equations. We show that for a two-species competition model it is possible that two species spread at different speeds, and there exists a traveling wave solution. For an $m$-species competition model, under the assumption that species have the same dispersal and growth properties but have different competition abilities, we establish the existence of traveling wave solutions.

Effects of time delay on symmetric two-species competition subject to noise

Noise and time delay act simultaneously on real ecological systems. The Lotka-Volterra model of symmetric two-species competition with noise and time delay was investigated in this paper. By means of stochastic simulation, we find that (i) the time delay induces the densities of the two species to periodically oscillate synchronously; (ii) the stationary probability distribution function of the two-species densities exhibits a transition from multiple to single stability as the delay time increases; (iii) the characteristic correlation time for the sum of the two-species densities squared exhibits a nonmonotonic behavior as a function of delay time. Our results have the implication that the combination of noise and time delay could provide an efficient tool for understanding real ecological systems.

Periodic solutions of a discrete two-species competitive model with stage structure

In this paper, we investigate the following discrete periodic stage-structure model. x 1 ( k + 1 ) = x 1 ( k ) exp { − a 1 ( k ) + b 1 ( k ) x 2 ( k ) x 1 ( k ) } , x 2 ( k + 1 ) = x 2 ( k ) exp { a 2 ( k ) x 1 ( k ) x 2 ( k ) − b 2 ( k ) − c ( k ) x 2 ( k ) − β 1 ( k ) x 3 ( k ) } , x 3 ( k + 1 ) = x 3 ( k ) exp { d ( k ) − e ( k ) x 3 ( k ) − β 2 ( k ) x 2 ( k ) } . The sufficient and realistic conditions are obtained for the existence of a positive periodic solution of this system.

Multiple mixed-type attractors in a competition model

We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in which both species are present) and exclusion attractors (in which one species is absent). Recent studies have investigated the inclusion of life-cycle stages in competition models as a casual mechanism for the existence of these kinds of multiple attractors. In this paper we investigate the role of nonlinearities in competition models without life-cycle stages.

Almost periodic solution of the non-autonomous two-species competitive model with stage structure

Sufficient conditions are obtained for the existence of a unique, globally attractive, strictly positive (componentwise), almost periodic solution of a non-autonomous, almost periodic competitive two-species model with a stage structure in one species. An example together with its numeric simulations shows the feasibility of our main results. Our result generalizes the main results of Zeng et al. [G.Z. Zeng, L.S. Chen, L.H. Sun, Y. Liu, Permanence and the existence of the periodic solution of the non-autonomous two-species competitive model with stage structure, Adv. Complex Syst., 7 (3&4) (2004) 385–393]