Lotka-Volterra Competition Model Research Articles

Bifurcation analysis of ecological and evolutionary processes in ecosystems

Bifurcation theory is commonly used to study the dynamical behaviour of ecosystems. It involves the analysis of points in the parameter space where the stability of the system changes qualitatively. Generally, such changes are related only to changes in environmental parameters, while the organism’s trait values are assumed to be constant. In reality, however, these trait values also change, though on a longer (evolutionary) time scale. On an ecological time scale, such evolutionary changes often come down to mutants invading a resident population. Points in the trait space where invasibility changes correspond to transcritical bifurcations. Therefore, bifurcation theory may also be used to study the evolutionary dynamics of ecosystems. First, the bifurctation approach is explained by analyzing a simple and well known Lotka–Volterra competition model where the competition coefficients and the carrying capacity are trait-dependent. However, the bifurcation approach is especially suitable for the analysis of more complex models. Therefore, the advantages and differences with the Adaptive Dynamics (AD) approach are discussed in more detail by means of the analysis of a more realistic and complex ecosystem model consisting of mixotrophic organisms.

Mixotrophy, allelopathy and the population dynamics of phagotrophic algae (cryptophytes) in the Darss Zingst Bodden estuary, southern Baltic

Seasonal plankton dynamics in the Darss Zingst Bodden Chain (DZBC), a southern Baltic estuary, are characterised by summer cyanophyte/chlorophyte dominance, but with winter blooms of mixotrophic cryptophytes, particularly associated with periods of ice cover. Analysis and simulation of a mixotrophic Lotka-Volterra competition model shows that coexistence of algae and cryptophytes is unlikely under static conditions, with seasonal or stochastic variation in parameters, as well as with time lags. However, incorporating production by cyanophytes/chlorophytes of non- toxic allelopathic substances allows stable coexistence of algae and mixotrophs. When parameterised using empirical DZBC data for light, temperature and ice cover, the model gives a good represen- tation of cryptophyte dynamics and bloom formation. We conclude that both mixotrophy and allelopathy are necessary ingredients of community coexistence and dynamics in the DZBC.

Overyielding and stable species coexistence

The concept of overyielding originated in plant sciences in the 1950s and 1960s and was widely used in the following decades to assess whether mixtures of plants performed better than expected when compared with monocultures. Overyielding has re-emerged in the last few years as an important method in the analysis of biodiversity experiments (Hector, 1998; Loreau, 1998; Loreau et al., 2001, 2002; Hooper et al., 2005) and other new research areas (Bernasconi et al., 2003). Biodiversity experiments manipulate community diversity (while holding other factors constant) to investigate impacts on ecosystem functioning. Previously, use of the overyielding concept has been limited mainly to the analysis of community ecology experiments on species interactions and in agricultural research, particularly intercropping. However, there has been relatively little work that assesses the overyielding concept in the context of community ecology theory. Loreau (2004) used the classical Lotka–Volterra competition model to investigate overyielding and functional redundancy of species in the context of theory on the stable coexistence of species (Fig. 1). In this issue, Beckage & Gross (pp. 140–148) also use Lotka–Volterra competition models to assess the frequency and degree of overyielding of theoretical communities.

The linear and nonlinear diffusion of the competitive Lotka–Volterra model

This paper has studied the effects of linear and nonlinear diffusion of the competitive Lotka–Volterra model, and has investigated how the linear and nonlinear diffusions lead from the extinction of one species to the persistence or global asymptotic stability of all species. This research has important implications in the design of nature reserves.

Loops and branches of coexistence states in a Lotka–Volterra competition model

A two-species Lotka–Volterra competition–diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.

Complex spatiotemporal dynamics in Lotka–Volterra ring systems

Mathematical models in ecology often need to incorporate spatial dependence to accurately model real-world systems. We consider competitive Lotka–Volterra models modified to include this spatial dependence through organization of the competing species into a one-dimensional ring by an appropriate choice of the interaction matrix. We show that these systems can exhibit complex dynamics, spatiotemporal chaos, and spontaneous symmetry breaking. A high-dimensional, spatially homogeneous, nearest-neighbor example with interaction strengths decreasing with distance is characterized including an analysis of how the dynamics of the system vary with dimension. We also show the existence of Lyapunov functions that arise from this method of including spatial dependence and how they prohibit complex dynamics for certain regions of the parameter space. We utilize these Lyapunov functions to reduce the required calculation time in brute-force searches of parameter space. A short comparison is given to derived line systems including a contrast between the eigenvalues of the two systems. .

Bifurcation structure of positive stationary solutions for a Lotka–Volterra competition model with diffusion I: Numerical verification of local structure

This paper is concerned with the bifurcation structure of positive stationary solutions for a generalized Lotka–Volterra competition model with diffusion. To establish the structure, the bifurcation theory and the interval arithmetic are employed.

Spatial coexistence of phytoplankton species in ecological timescale

AbstractThe species diversity of phytoplankton is usually very high in wild aquatic systems, as seen in the paradox of plankton. Coexistence of many competitive phytoplankton species is extremely common in nature. However, experiments and mathematical theories show that interspecific competition often leads to the extinction of most inferior species. Here, we present a lattice version of a multi‐species Lotka–Volterra competition model to demonstrate the importance of local interaction. Its mathematical equilibrium is the exclusion of all but one superior species. However, temporal coexistence of many competitive species is possible in an ecological time scale if interactions are local instead of global. This implies that the time scale is elongated many orders when interactions are local. Extremely high species diversity of phytoplankton in aquatic systems may be maintained by spatial coexistence in an ecological time scale.

Permanence of a delayed non-autonomous Gilpin–Ayala competition model

In this paper, we consider a delayed non-autonomous n-species Gilpin–Ayala competitive system, which is more general and more realistic then classical Lotka–Volterra competition model. By means of Ahmad and Lazer’s definitions of lower and upper averages of a function, we give the average conditions for the permanence of the system. It is shown that our result is the generalization of those of Zhao et al. [J.D. Zhao, J.F. Jiang, A.C. Lazer, The permanence and global attractivity in a nonautonomous Lotka–Volterra system, Nonlinear Analysis: Real World Applications, 5 (4) (2004), 265–276]. Our results also supplement the results of Fan and Wang [M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin–Ayala competition model, Computer and Mathematics with Applications, 40 (2000), 1141–1151].

Modelling responses of pine savannas to climate change and large-scale disturbance

Global warming can potentially influence ecological communities through altered disturbance regimes in addition to increased temperatures. We investigate the response of pine savannas in the southeastern United States to global warming using a simple Lotka-Volterra competition model together with predicted changes to fire and hurricane disturbance regimes with global climate change. In the southeastern United States, decreased frequency of both fires and hurricanes with global warming will shift pine savannas toward a forested state. A CO2 fertilization effect that increases the growth rate of tree populations will also push southeastern landscapes from open savannas towards closed forests. Transient dynamics associated with climate driven changes in vegetation will last on the order of decades to a century. In our model, the sensitivity of savannas to relative changes in the frequency of fire versus hurricanes is linearly dependent on the growth rate and mortality of trees in fire and hurricane disturbances.

Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure

In this paper, we consider a Lotka-Volterra competition model with diffusion, and show that the global bifurcation structure of positive stationary solutions for the model is similar to that for a certain scalar reaction-diffusion equation. To do this, the comparison principle, the bifurcation theory, and the numerical verification are employed.

Permanence of nonautonomous Lotka–Volterra delay differential systems

In this work, applying the results offered by S. Ahmad and A.C. Lazer [On a property of nonautonomous Lotka–Volterra competition model, Nonlinear Anal. 37 (1999) 603–611] and the recent work of R. Redheffer [Mean values and the nonautonomous May–Leonald equations, Nonlinear Anal. Real World Appl. 4 (2003) 301–306] to an nonautonomous Lotka–Volterra differential system with finite delays, we establish sufficient conditions for the permanence of the system.

Average conditions for permanence and extinction in nonautonomous Gilpin–Ayala competition model

In this paper, we consider a general nonautonomous n-species Gilpin–Ayala competitive system, which is more general and more realistic than classical Lotka–Volterra competition model. By means of Ahmad and Lazer's definitions of lower and upper averages of a function, we first give the average conditions for the permanence and global attractivity of the system. Next, for each r ⩽ n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n - r species are driven to extinction. Examples show the feasibility of the main results.

On the effects of migration and spatial heterogeneity on single and multiple species

We first investigate in a logistic model the effects of migration and spatial heterogeneity of the environment on the total population size at equilibrium of a single species. Our study shows that (i) the total population size is maximized at some intermediate migration rate, and hence is a non-monotone function of the migration rate; (ii) heterogeneity of the environment increases the population size. In the second part of this paper, these findings are applied to ecological invasions. For a two-species Lotka–Volterra competition model with migration, we show that (i) without migration, the invading species eliminates the resident species at every point of the habitat, whereas when migration is present, for certain ranges of migration rates the invader may be eliminated when it is rare; and (ii) without migration, the two species can coexist at every point of the habitat, whereas when migration is present, for some ranges of migration rates one of the species is extinguished for all initial conditions.

A nonstandard discretization method for Lotka–Volterra models that preserves periodic solutions

Three discretization schemes are applied to the Lotka–Volterra competition models with an interior fixed point. The local dynamics of these discrete-time models near the interior fixed point are analysed. One of the discrete models shows dynamical consistency with the continuous model in the sense that it produces periodic solutions when the continuous model does posses periodic solutions while the other two discrete models produce solutions which either spiral into or out of the fixed point. The discrete model derived from Kahan's discretization method preserves Hopf bifurcations in the continuous model.

Competitive coexistence of two Pneumocystis species

Pneumocystis are fungal pathogens of mammalian lungs that can cause lethal pneumonia in immunocompromised hosts. In some mammals, coinfections of genetically distinct Pneumocystis populations have been identified, but the nature of their interaction and its significance are unknown. Two species that infect rats, Pneumocystis carinii and Pneumocystis wakefieldiae, were studied over a 6-year period, representing approximately 700 generations of Pneumocystis. Population densities of each species were analyzed within the framework of the Lotka–Volterra competition model, which revealed the two species were in competition and predicted competitive exclusion of one species. However, stable coexistence was observed in 460 replicate populations. Selected extrinsic factors that might mitigate the extinction were evaluated. Logistic-regression analyses showed that higher relative humidity and higher organism lung burdens were associated with infections comprised of P. carinii alone, while lower temperatures and an increased rat census were associated with the presence of P. wakefieldiae. PCR and immunofluorescent analysis of rat lung tissue showed that both species were present within the same alveoli, excluding habitat heterogeneity as a mechanism of coexistence. These data suggest that P. carinii and P. wakefieldiae were in competitive coexistence, which was influenced in part by extrinsic factors. To our knowledge, this is the first report to evaluate interactions of pathogenic fungal species within a mammalian host using ecological models.

Modelling and analysis of a competitive model with stage structure

A two-species Lotka-Volterra type competition model with stage structures for both species is proposed and investigated. In our model, the individuals of each species are classified as belonging either the immature or the mature. First, we consider the stage-structured model with constant coefficients. By constructing suitable Lyapunov functions, sufficient conditions are derived for the global stability of nonnegative equilibria of the proposed model. It is shown that three typical dynamical behaviors (coexistence, bistability, dominance) are possible in stage-structured competition model. Next, we consider the stage-structured competitive model in which the coefficients are assumed to be positively continuous periodic functions. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are obtained for the existence of positive periodic solutions to the model. Numerical simulations are also presented to illustrate the feasibility of our main results.

On the impossibility of coexistence of infinitely many strategies

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.

Two species competition with an inhibitor involved

The dynamics of the solution flow of a two-species Lotka-Volterra competition model with an extra equation for simple inhibitor dynamics is investigated. The model fits into the abstract framework of two-species competition systems (or $K$-monotone systems), but the equilibrium representing the extinction of both species is not a repeller. This feature distinguishes our problem from the case of classical two-species competition without inhibitor (classical case for short), where a basic assumption requires that equilibrium to be a repeller. Nevertheless, several results similar to those in the classical case, such as competitive exclusion and the existence of a "thin" separatrix, are obtained, but differently from the classical case, coexistence of the two species or extinction of one of them may depend on the initial conditions. As in almost all two species competition models, the strong monotonicity of the flow (with respect to a certain order on $\mathbb R^3$) is a key ingredient for establishing the main results of the paper.

A competition model for two CPU vendors

In a severely competing economic environment, the competing ability of a company must be improved continuously as the reaction to the outer competition pressure. We propose a model developed from Lotka–Volterra competition model with time dependent parameters other than the equilibrium theory so as to describe some characteristics of the technology innovation. The time-dependent parameters comprise carrying capacities and competitive effects. We assume that the technological index is represented, in some degree, by the highest CPU clock frequency. We have quantitatively studied quarterly revenues of AMD and Intel, two chief vendors of the central processing unit (CPU). Moreover, we give the empirical values of the basic parameter set according to theoretical analysis and our simulation results fit the revenue data with reasonable agreement. It demonstrates that the model is capable of describing some important commercial phenomena in certain technology-leading industries. The technology innovation but not the strategy, is the crucial factor of competition, and the first-mover advantage will not be always unbroken. Furthermore, we have found that the unwilling mutualism appeared in the present model cannot be explained by the strategic behavior theory.