Stability analysis of predator-prey models via the liapunov method

As is well-known from the classical applications in the electrical and mechanical sciences, energy is a suitable Liapunov function: thus, by analogy, all energy functions proposed in ecology are potential Liapunov functions. In this paper, a generalized Lotka-Volterra model is considered and the stability properties of its non-trivial equilibrium are studied by means of an energy function first proposed by Volterra in the context of conservative ecosystems. The advantage of this Liapunov function with respect to the one that can be induced through linearization is also illustrated.     

Liapunov functions: Geometry and stability

Summary Several geometrical interpretations of Liapunov functions for non-linear ecological models are examined and their limitations pointed out. In particular the geometrical nonuniqueness of Liapunov functions is illustrated by displaying explicitly and comparing four different Liapunov functions for the symmetric competition model for two species. The main point is that considerable care must be taken in using the geometrical properties of an arbitrary Liapunov function as a guide to stability under perturbations.     

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.     

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models.     

On global stability of discrete population models

A necessary and sufficient condition for the global stability of a large class of discrete population models is provided which does not require the construction of a Liapunov function. The general result is applied to difference equations defined in terms of “two hump” functions and to an example of frequency dependent selection.     

Estimating the effect of continual disturbances on discrete-time population models

In this paper, methods for estimating the effect of continual (but bounded) disturbances on ecosystems modelled by difference equations are discussed. The approach adopted is to estimate the region of state space (called a reachable set) which can be reached by the disturbed system from an initial healthy state in a given time period. Liapunov stability methods for estimating these reachable sets are presented and applied to two specific population models.     

Modeling arbuscular mycorrhizal infection: is % infection an appropriate variable?

Several available models of arbuscular mycorrhizal infection are based on fitting % infection to a logistic curve and then relating the various parameters to biological functions. I suggest here that this direction is misleading. Percent infection is a value derived from the growth of two interdependent but distinct organisms, each of which is seeking to maximize its own growth and survival. I suggest that two-organism models, such as those derived from Lotka-Volterra equations, are more useful for understanding the biology and functioning of mycorrhizae. Accepted: 22 October 2000     

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.     

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。     

When the feasibility of an ecosystem is sufficient for global stability?

We show via a Liapunov function that in every model ecosystem governed by generalized Lotka-Volterra equations, a feasible steady state is globally asymptotically stable if the number of interaction branches equals n-1, where n is the number of species. This means that the representative graph for which the theorem holds is a 'tree' and not only an alimentary chain. Our result is valid also in the case of non-homogeneous systems, which model situations in which input fluxes are present.     

Asymptotic behaviour of the non-autonomous competing two-species Lotka-Volterra models with impulsive effect

In this paper, the nonautonomous competing two-species Lotka-Volterra models with impulsive effect are considered, where all the parameters are time-dependent and asymptotically approach the corresponding periodic functions. Under some conditions, it is shown that the semi-trivial positive solutions of the models asymptotically approach the semi-trivial positive periodic solutions of the corresponding periodic system. It is also shown that the positive solution of the models asymptotically approach the positive periodic solution of the corresponding periodic system.     

具有三个年龄阶段的单种群自食模型

建立并研究了两个具有三个年龄阶段的单种群自食模型．这篇文章的主要目的是研究时滞对种群生长的作用,对于没有时滞的的模型,我们利用Liapunov函数,得到了系统平衡点全局渐近稳定的充分条件;而具有时滞的的模型,我们得到,随着时滞T增加,当系数满足一定条件时,正平衡点的稳定性可以改变有限次,最后变成不稳定;否则,时滞模型的正平衡点的稳定性不改变。     

Non-equilibrium thermodynamics of temporally oscillating chemical reactions.

Two thermodynamic quantities are introduced: the entropy change due to a variation of chemical affinity from a steady, state. to some other state and the corresponding entropy production. The entropy change is always negative definite except at the steady state and is capable of being a Liapunov function. The phase-plane behaviour of the entropy production along the trajectory generated by kinetic equations is investigated in connection with the stability of steady state. The examples taken on that occasion are the Volterra-Lotka and Prigogine-Lefever models. The non-equilibrium thermodynamic properties common to the oscillating reactions in two-variable system are in general considered with an emphasis on the thermodynamic analysis for the direction of rotation of the trajectory generated by the two-variable kinetic equations.     

Niche overlap and invasion of competitors in random environments I. Models without demographic stochasticity

The relationship between persistent, small to moderate levels of random environmental fluctuations and limits to the similarity of competing species is studied. The analytical theory hinges on deriving conditions under which a rare invading species will tend to increase when faced with an array of resident competitors in a fluctuating environment. A general approximation scheme predicts that the effects of low levels of stochasticity will typically be small. The technique is applied explicitly to a class of symmetric, discrete-time stochastic analogs of the Lotka-Volterra equations that incorporate cross-correlation but no autocorrelation. The random environment limits to similarity are always very close to the corresponding constant environment limits. However, stochasticity can either facilitate or hinder invasion. The exact limits to similarity are extremely model-dependent. In addition to the symmetric models, an analytically tractable class of models is presented that incorporates both auto- and cross-correlation and no symmetry assumptions. For all of the models investigated, the analytical theory predicts that small-scale stochasticity does little, if anything, to limit similarity. Extensive Monte Carlo results are presented that confirm the analytical results whenever the dynamics of the discretetime models are biologically reasonable in the sense that trajectories do not exhibit unrealistic crashes. Interestingly, the class of stochastic models that is well behaved in this sense includes models whose deterministic analogs are chaotic. The qualitative conclusion, supported by both the analytical and simulation results, is that for competitive guilds adequately modeled by Lotka-Volterra equations including small to moderate levels of random fluctuations, practical limits to similarity can be obtained by ignoring the stochastic terms and performing a deterministic analysis. The mathematical and biological robustness of this conclusion is discussed.     

Nonvulnerability of ecosystems in unpredictable environments

One way to bracket the effects of a real environment on an ecosystem during a finite time interval is to use the concept of vulnerability. If a deterministic model ecosystem has a good Lyapunov function, it may be possible to derive simple and useful tests for the system to be nonvulnerable. For a subset of Lotka-Volterra models, the system is nonvulnerable if the smallest eigenvalue of a certain matrix is not only positive, but is greater than a positive number, which depends on a priori estimates for the bounds on the unpredictable forcing functions. The bounded but unknown functions which act on the Lotka-Volterra equations also can be interpreted as errors in the system's equations which can be tolerated without a qualitative change in the behaviour of its solutions.     

Diffusion effect on stability of Lotka-Volterra models

This paper analyses the diffusion effect on stability in Lotka-Volterra systems for a patch-type environment. Applying the extended stability theorem of LaSalle, some classes of patches for which the diffusion does not affect the system's stability are drawn. Further, complicated dynamical behaviours in two-prey, one-predator diffusion models are given when the patch does not belong to the above classes. This research was supported by the Ministry of Education, Science and Culture, Japan, under Grant AEYS 60740109.     

The Tangled Bank: The maintenance of sexual reproduction through competitive interactions

The maintenance of sexual reproduction is discussed using a model based on the familiar Lotka-Volterra competition equations. Both the equilibrium and the stability conditions that allow a sexual population to resist invasion by a single asexual clone are considered. The equilibrium conditions give results similar to previous models: When the cost of sex, within phenotype niche width, and environmental variance are low, the sexual population coexists with the asexual clone and remains at a high density. However, the asexual clone is never completely excluded. Analysis of the stability conditions shows a different picture: The introduction of an asexual clone considerably reduces the stability of the community. However, owing to its larger total niche width, the sexual population exists partly in a “competitor-free space” where the asexual clone has almost no influence on the outcome of the interactions. Therefore the asexual clone is less stable than the sexual population and has a higher probability of extinction. In contrast, the sexual population does not become extinct, since the extreme phenotypes remain at a stable, though low, density, and the central phenotypes, where stability is low, are recreated every generation through recombination. I therefore conclude that the ecological conditions under which sexual reproduction is favored over asexual reproduction are fairly easily attained and are more general than previous analyses had suggested.     

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.