种间竞争的一个新的数学模型—对经典的Lotka—Vol—terra竞争方程的扩充

本文根据营养动力学理论,建立了一类种间竞争的新的数学模型:它是单种群增长的Cui-Lawson模型,在种间竞争上的推广。新的种间竞争模型克服了经典的种间竞争的Lotka-Volteira方程的局限与不足,具有更广泛和复杂的行为,并在特殊条件下以Lotka-Volterra竞争方程为其特例。因此,新的种间竞争的数学模型是更一般的解释性模型,是对经典的Lotka-Voterra竞争方程的扩充。     

Convergence in a resource-based competition system

A resource-based competition model of two consumer species and one resource species is formulated in the form of a Lotka-Volterra system. The competition involves both exploitation and interference. By a method of asymptotic estimates, sufficient conditions are derived for the three species system to converge ast→∞ to an equilibrium point with all three species present; a generalization of the result forn≥2 and single resource species is indicated. The strong form of equilibrium perisistence of the three species consumer-resource system is achieved by the ability of each of the consumer species to exploit the resource and interfere with others in such a way which will avoid exclusion by the other.     

Biodiversity,habitat area,resource growth rate and interference competition

For the majority of species, per capita growth rate correlates negatively with population density. Although the popular logistic equation for the growth of a single species incorporates this intraspecific competition, multi-trophic models often ignore self-limitation of the consumers. Instead, these models often assume that the predator-prey interactions are purely exploitative, employing simple Lotka-Volterra forms in which consumer species lack intraspecific competition terms. Here we show that intraspecific interference competition can account for the stable coexistence of many consumer species on a single resource in a homogeneous environment. In addition, our work suggests a potential mechanism for field observations demonstrating that habitat area and resource productivity strongly positively correlate to biodiversity. In the special case of a modified Lotka-Volterra model describing multiple predators competing for a single resource, we present an ordering procedure that determines the deterministic fate of each specific consumer. Moreover, we find that the growth rate of a resource species is proportional to the maximum number of consumer species that resource can support. In the limiting case, when the resource growth rate is infinite, a model with intraspecific interference reduces to the conventional Lotka-Volterra competition model where there can be an unlimited number of coexisting consumers. This highlights the crucial role that resource growth rates may play in promoting coexistence of consumer species.     

On a resource based ecological competition model with interference

A resource based ecological competition model with interference is proposed. The model is based on Lotka-Volterra dynamics with two predators competing for a single, limited prey. Interference effects are considered in this article. When the interference coefficient, expressing the damage effect from its rival, is small, the mathematical analysis shows that the winner in purely exploitative competition still outcompetes its rival. However, if the interference coefficient is large enough then the competition outcome will depend on initial population of predator species.     

Effective competition determines the global stability of model ecosystems

We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.     

A model of competition

Summary The paper considers a model of competition, based upon the Lotka-Volterra equations, which explicitly considers the effect of density independent mortality upon the outcome of competition. The model's possible application to wild Drosophila species in Europe are considered.     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

Competitive overlap and coexistence

The possibility of equilibrium is studied for model assemblages of competing species and their resources. The “assemblage niche” is defined as the set of resource productivities which yields an equilibrium population exceeding zero for all species. A radius of this set, which is a measure of the ability of the assemblage to have equilibrium states, is defined and estimated. This radius decreases as resource utilization overlap increases; the behavior is compared with known results concerning response to rapid resource fluctuations. A system of ordinary differential equations having such an equilibrium is studied. It is shown that a global asymptotic stability property holds in regions with boundaries defined by a certain scalar function, if the specific productivity satisfies a monotonicity condition. This generalizes known results, which have been obtained for antisymmetric Lotka-Volterra systems.     

Effects of intraguild predation on resource competition.

In this work, a simple Lotka-Volterra model of intraguild predation with three species is analysed, searching for the effect of the top predator on the coexistence with its prey-competitor species. Apart from the well-known result that the intraguild prey must be superior in the competition for the shared prey in order to make coexistence possible, the magnitude of intraguild predation and the form by which the intraguild predator makes use of the intraguild prey have important consequences upon the dynamics, extending or restricting the possibilities of coexistence. These results are easily obtained by nullcline analysis. Also, some interesting results are obtained for the same model but including saturating functional response.     

A note on exact solutions of two prey-predator equations

Exact solutions are obtained and discussed for classes of Lotka-Volterra and Leslie-Gower systems governing the interaction of two species. The classes are defined by certain constraints which are imposed on the time-dependent parameters of the equations. A general result for such systems is that each species is characterised by two time-scales: one representing natural growth and the other, the interdependence of the species.     

A COEVOLUTIONARY ISOMORPHISM APPLIED TO LABORATORY STUDIES OF COMPETITION

Some empirical consequences of an isomorphism between the Lotka-Volterra competitive model and a coevolutionary competitive model are developed. In both the Lotka-Volterra and coevolutionary models, four competitive outcomes are possible: 1) species one wins, 2) species two wins, 3) indeterminate outcome, and 4) stable coexistence. These two models are isomorphic in the sense that the inequalities associated with a particular competitive outcome of the Lotka-Volterra model correspond in a one-to-one manner with similar inequalities associated with the same competitive outcome of the coevolutionary model. The inequalities of the Lotka-Volterra model involve the competition coefficients themselves, while the inequalities of the coevolutionary model involve the genetic variances and covariances of the competition coefficients. The isomorphism suggests some alternative interpretations of the results of classical laboratory studies of competition. The Lotka-Volterra (or ecological) hypotheses postulate that the competition coefficients are constant and that genetic considerations play no role in determining the competitive outcome. By contrast, the evolutionary hypotheses derived from the coevolutionary model postulate that the competition coefficients are variables and that the genetic variances and covariances of the competition coefficients determine the competitive outcome. The isomorphism is applied to competitive exclusion and coexistence, and to competitive indeterminacy in Tribolium. In particular, the evolutionary hypotheses isomorphic to the two classical explanations of competitive indeterminacy, the demographic stochasticity and genetic founder effect hypotheses, are constructed. The theory developed here and in a previous paper (Pease, 1984) provides one perspective on the relation among the Lotka-Volterra competition theory, quantitative genetics, competitive exclusion, the reversal of competitive dominance, coexistence, competitive indeterminacy in Tribolium, and experiments investigating the relation between genetic variability and the rate of evolution of fitness.     

An alternative to Lotka-Volterra competition in coarse-grained environments

Competition in a coarse-grained heterogeneous environment is considered. It is assumed that each individual adjust has position within the macrohabitat so as to try to maximize a utility whose value is a function of position within the habitat and of the population density at that position. Robust qualitative conclusions concerning the resulting population behavior relate niche breadth, niche shift, and resource partitioning to changes in population numbers. In particular populations avoid competition by reducing coarse-grained niche overlap so that overlap is a measure not of the degree of competition but of its absence. Specific models of exploitation or interference competition give specific quantitative results which are summarized. Competition equations are derived from underlying models which imply Lotka-Volterra equations in a fine-grained environment (MacArthur, R., and Levins, R. 1967. Amer. Natur.101, 377–385) but which are nonlinear in per capita growth in a coarse-grained one. The findings are relevant to theoretical studies of competition, limiting similarity, species packing, and the evolution of niche position as well as to practical problems of data analysis and resource management. The extension to predator-prey interactions is outlined.     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

Predictive modeling of mixed microbial populations in food products: evaluation of two-species models

Predictive microbiology is an emerging research domain in which biological and mathematical knowledge is combined to develop models for the prediction of microbial proliferation in foods. To provide accurate predictions, models must incorporate essential factors controlling microbial growth. Current models often take into account environmental conditions such as temperature, pH and water activity. One factor which has not been included in many models is the influence of a background microflora, which brings along microbial interactions. The present research explores the potential of autonomous continuous-time/two-species models to describe mixed population growth in foods. A set of four basic requirements, which a model should satisfy to be of use for this particular application, is specified. Further, a number of models originating from research fields outside predictive microbiology, but all dealing with interacting species, are evaluated with respect to the formulated model requirements by means of both graphical and analytical techniques. The analysis reveals that of the investigated models, the classical Lotka-Volterra model for two species in competition and several extensions of this model fulfill three of the four requirements. However, none of the models is in agreement with all requirements. Moreover, from the analytical approach, it is clear that the development of a model satisfying all requirements, within a framework of two autonomous differential equations, is not straightforward. Therefore, a novel prototype model structure, extending the Lotka-Volterra model with two differential equations describing two additional state variables, is proposed to describe mixed microbial populations in foods.     

Evolutionary disarmament in interspecific competition.

Competitive asymmetry, which is the advantage of having a larger body or stronger weaponry than a contestant, drives spectacular evolutionary arms races in intraspecific competition. Similar asymmetries are well documented in interspecific competition, yet they seldom lead to exaggerated traits. Here we demonstrate that two species with substantially different size may undergo parallel coevolution towards a smaller size under the same ecological conditions where a single species would exhibit an evolutionary arms race. We show that disarmament occurs for a wide range of parameters in an ecologically explicit model of competition for a single shared resource; disarmament also occurs in a simple Lotka-Volterra competition model. A key property of both models is the interplay between evolutionary dynamics and population density. The mechanism does not rely on very specific features of the model. Thus, evolutionary disarmament may be widespread and may help to explain the lack of interspecific arms races.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.