Mathematical models for continuous culture growth dynamics of mixed populations subsisting on a heterogeneous resource Base: I. Simple competition

The classical Monod model for bacterial growth in a chemostat, based on a Michaelis-Menten kinetic analog, is restated in terms of an approximate Lotka-Volterra formulation. The parameters of these two formulations are explicitly related; the new model is easier to work with, but yields the same results as the original. The model is then extended to the case where multiple alternate substrates may be growth limiting, using the corresponding kinetic analogs for multiple-substrate enzymes. Again, one is led to a Lotka-Volterra analog. In the multiple-substrate model, however, coexistence of multiple genotypes is possible, in contrast to the single-substrate model. The usual Lotka-Volterra conditions for existence and stability of pure or mixed equilibria may all be translated into corresponding statements about the parameters of the chemostat system. Possible extensions to deal with metabolic inhibition, cross-feeding, and predation are indicated.     

普生轮藻浸提液对两种淡水藻类的化感抑制作用及其数学模型

利用大型水生植物的化感作用抑制水华藻类是水域生态学研究的热点课题之一。探讨了不同浓度普生轮藻浸提液对产毒铜绿微囊藻和斜生栅藻(单纯以及混合藻类)的抑制作用,并根据实验过程中得到的数据和数据特征,在传统的Logistic模型和Lotka-Volterra模型基础上,通过微元法建立了普生轮藻浸提液对单纯产毒铜绿微囊藻、单纯斜生栅藻抑制的数学模型以及两藻混合时抑制的数学模型。结果表明,(1)普生轮藻浸提液无论对单独的毒性铜绿微囊藻或斜生栅藻还是共生状态的毒性铜绿微囊藻和斜生栅藻均有很强抑制作用,且对毒性铜绿微囊藻的抑制作用要显著高于对斜生栅藻;(2)所建立的抑藻模型可有效表征和预测在一定范围内的产毒铜绿微囊藻、斜生栅藻及其混合藻在普生轮藻浸提液胁迫下藻密度随时间变化的规律;通过这些模型可方便地计算出实验期间任何时间节点上普生轮藻浸提液的半抑制浓度(EC50)、最小有效浓度(MIC)等指标的预测值、混合藻在小生境中相对稳定时的预测值等等。该研究可为实际抑藻的方案制定和实施提供有价值的数据支撑和参考,具有一定的理论与应用意义。     

The effects of interference competition on stability,structure and invasion of a multi-species system

Summary An interference competition model for a many species system is presented, based on Lotka-Volterra equations in which some restrictions are imposed on the parameters. The competition coefficients of the Lotka-Volterra equations are assumed to be expressed as products of two factors: the intrinsic interference to other individuals and the defensive ability against such interference. All the equilibrium points of the model are obtained explicitly in terms of its parameters, and these equilibria are classified according to the concept of sector stability. Thus survival or extinction of species at a stable equilibrium point can be determined analytically.The result of the analysis is extended to the successional processes of a community. A criterion for invasion of a new species is obtained and it is also shown that there are some characteristic quantities which show directional changes as succession proceeds.     

Predator-Prey Interactions of Dictyostelium discoideum and Escherichia coli in Continuous Culture

Dictyostelium discoideum and Escherichia coli were aerobically propagated in mixed continuous culture in a predator-prey relationship, and the effects of temperature and holding times were examined. Oscillations developed in the concentration of glucose, the limiting substrate for E. coli, and in the densities of the two populations, but eventually steady-state populations were reached. The experimental data were analyzed according to the Lotka-Volterra model for prey-predator relationships and by the Monod model for saturation kinetics. A comparison of the adequacy of the two models in describing predation is given.     

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.     

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.     

A solution to the accelerated-predator-satiety Lotka-Volterra predator-prey problem using Boubaker polynomial expansion scheme

In this study, an analytical method is introduced for the identification of predator-prey populations time-dependent evolution in a Lotka-Volterra predator-prey model which takes into account the concept of accelerated-predator-satiety.Oppositely to most of the predator-prey problem models, the actual model does not suppose that the predation is strictly proportional to the prey density. In reference to some recent experimental results and particularly to the conclusions of May (1973) about predators which are ‘never not hungry’, an accelerated satiety function is matched with the initial conventional equations. Solutions are plotted and compared to some relevant ones. The obtained trends are in good agreement with many standard Lotka-Volterra solutions except for the asymptotic behaviour.     

Invariant distributions for multi-population models in random environments

We obtain the existence of a solution and invariant distribution for systems of stochastic differential equations which represent populations in random environments. The method used is a stochastic Lyapunov function, based on a theorem of Kushner. The method is applied to a system of two populations exchainging individuals through migration, and to a generalized n-dimensional Lotka-Volterra system.     

基于后推法的n维Lotka-Volterra竞争模型的稳定化设计

通过后推设计方法,研究一类两两相互竞争的n堆Lotka-Volterra模型的全局稳定化问题．在状态反馈控制下,获得了使闭环系统在正平衡点处全局渐近稳定的控制律．     

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c^* dividing the waves into two types, waves of speed c < c^* being one type, waves of speed c ? c^* being of the other type. We present numerical evidence that for this system the wave of speed c^* is stable, and that c^* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.     

Application of perturbation theory to the nonlinear Volterra-Gause-Witt model for prey-predator interaction

Krylov-Bogoliubov-Mitropolsky perturbation method was used to study the effect of nonlinearity in the Volterra-gause-Witt (VGW) model for a two species prey-predator system. The first order corrections to both the frequency of oscillation and the amplitude of the linearized system were computed. It was found that the basic qualitative features of the nonlinearity are exhibited by the first order result. We have also discussed the Lotka-Volterra problem which is a special case of VGW model.     

Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment

The dynamics of single populations up to ecosystems, are often described by one or a set of non-linear ordinary differential equations. In this paper we review the use of bifurcation theory to analyse these non-linear dynamical systems. Bifurcation analysis gives regimes in the parameter space with quantitatively different asymptotic dynamic behaviour of the system. In small-scale systems the underlying models for the populations and their interaction are simple Lotka-Volterra models or more elaborated models with more biological detail. The latter ones are more difficult to analyse, especially when the number of populations is large. Therefore for large-scale systems the Lotka-Volterra equations are still popular despite the limited realism. Various approaches are discussed in which the different time-scale of ecological and evolutionary biological processes are considered together.     

Competition between species: theoretical models and experimental tests

Experimental determinations of Drosophila population dynamics cannot be explained by the Lotka-Volterra model of interspecific competition. This paper presents other possible mathematical models of competition between species, and gives the results of experiments designed to test the validity of such models. Eight of the ten new models presented contain the Lotka-Volterra model as a special case. The experiments made to test the models are of two kinds. Type 1 experiments are continuous one- or two-species populations, which permit the estimation of the carrying capacity of each species and the numbers of the two species at the point of stable equilibrium. Type 2 experiments measure the change in numbers over a short time interval in populations started with many different initial densities of the two species. Type 2 experiments give information on the dynamics of the two-species system in the phase plane whose coordinates are the number of individuals of each species. The models accounting best for the results are models five and seven (Table II). Each of these two models contains one parameter more than the Lotka-Volterra model. Model five adds a nonlinear term of self-interaction (?β_iN²_i). Model seven has the form, dN_i/dt = r_iN/K^θ_ii(K^θ_ii ? N^θ_ii ? α_ijN_j^1?θ_ii). The exponential parameter θ removes the restriction of the logistic theory of population growth, that each individual added to the population decrease the rate of growth of the population by a constant amount. With model seven the rate of growth of a population of a single species need not have its maximum at $\text{K}\text{2}$, that is when the number of individuals is half the carrying capacity of the environment.     

基于反馈控制离散Lotka-Volterra竞争模型的持续生存和稳定性

针对一类具有偏离自变量的离散Lotka-Volterra竞争模型,考虑到不可避免的外界扰动,通过引入反馈控制,基于一定的分析技巧得到该系统持久性与全局稳定性的充分条件.生态意义上表明:在外界扰动下,具有偏离自变量的离散Lotka-Volterra竞争模型仍能持续生存并保持全局稳定发展.     

When the exception becomes the rule: The disappearance of limiting similarity in the Lotka-Volterra model

We investigate the transition between limiting similarity and coexistence of a continuum in the competitive Lotka-Volterra model. It is known that there exist exceptional cases in which, contrary to the limiting similarity expectation, all phenotypes coexist along a trait axis. Earlier studies established that the distance between surviving phenotypes is in the magnitude of the niche width 2σ provided that the carrying capacity curve differs from the exceptional one significantly enough. In this paper we studied the outcome of competition for small perturbations of the exceptional (Gaussian) carrying capacity. We found that the average distance between the surviving phenotypes goes to zero when the perturbation vanishes. The number of coexisting species in equilibrium is proportional to the negative logarithm of the perturbation. Nevertheless, the niche width provides a good order of magnitude for the distance between survivors if the perturbations are larger than 10%. Therefore, we conclude that limiting similarity is a good framework of biological thinking despite the lack of an absolute lower bound of similarity.     

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.     

Mathematical models for continuous culture growth dynamics of mixed populations subsisting on a heterogeneous resource base. II. Predation and trophic structure

Models are presented for the joint dynamics of predators and prey, maintained in continuous flow chemostat culture. The predators are visualized as subsisting on one or more prey organisms, which in turn are visualized as subsisting on one or more substrate resources supplied by the investigator. The dynamic equations are translated into an analogous Lotka-Volterra predation model, and the criteria for the existence and stability of various equilibria are indicated. Denoting the number of different predator organisms as N_H, the number of different prey organisms by N_I and the number of different substrates as N_J, it is shown that the joint coexistence of all components requires 0 ? N_I ? N_H ? N_J. The model is extended to more complex situations by including additional trophic layers and by allowing trophic layer “leap-frogging.” The model may always be translated into an approximately quadratic differential equation of the Lotka-Volterra type. The α- and β-coefficients of these latter are really variables, and become quite complex for some of the multi-layered models.