Gail S. K. Wolkowicz

Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates

A model of exploitative competition of n species in a chemostat for a single, essential, nonreproducing, growth-limiting resource is considered. S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760–763] applies LaSalle’s extension theorem of Lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by Michaelis–Menten dynamics. G. J. Butler and G. S. K. Wolkowicz [SIAM J. Appl. Math., 45 (1985), pp. 138–151], on the other hand, allow more general response functions (including monotone and nonmonotone functions), but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate, and hence can be ignored. By means of Lyapunov stability theory, the global dynamics of the model for a large class of response functions are studied, including both monotone and nonmonotone functions (though it is not as general as the class studied by Butler and Wolkowicz) and the results in ...

A MATHEMATICAL MODEL OF THE CHEMOSTAT WITH A GENERAL CLASS OF FUNCTIONS DESCRIBING NUTRIENT UPTAKE

A model of the chemostat involving n microorganisms competing for a single essential, growth-limiting substrate is considered. Instead of assuming the familiar Michaelis-Menten kinetics for nutrient uptake, a general class of functions is used which includes all monotone increasing uptake functions, but which also allows uptake functions that describe inhibition by the substrate at high concentrations.The qualitative behaviour of this generalized model is determined analytically. It is shown that the behaviour depends intimately upon certain parameters. Provided that all the parameters are distinct (which is a biologically reaonable assumption), at most one competitor survives. The substrate and the surviving competitor (if one exists), approach limiting values. Thus there is competitive exclusion. However, unlike the standard model, in certain cases the outcome is initial condition dependent.

BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSE ∗

We consider a predator-prey system with nonmonotonic functional response:

Predator-prey systems with group defence: the paradox of enrichment revisited

p(x)=\frac{mx}{ax^2+bx+1}

Bifurcation Analysis of a Predator-Prey System Involving Group Defence

. By allowing b to be negative (

Global asymptotic behavior of a chemostat model with discrete delays

b > -2\sqrt a

Competition in the chemostat: a distributed delay model and its global asymptotic behavior

), p(x) is concave up for small values of x > 0 a...

Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat

The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically).

Exploitative competition in a chemostat for two complementary, and possibly inhibitory, resources

A class of ODEs of generalized Gause type modeling predator-prey interaction is considered. The prey are assumed to exhibit a phenomenon called group defence, that is, predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase.Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a Hopf bifurcation. Conditions (that hold even in the case of no group defence) that ensure a subcritical Hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair (one stable and one unstable) of periodic orbits are given.Ecological ramifications are considered. Unlike the classical model, sufficient enrichment of the environment combined with group defence leads to extinction of the predator (deterministically) for almost all initial conditions, providing strong support fo...

Exploitative competition in the chemostat for two perfectly substitutable resources

This paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat. The model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations. In the case where only two species are engaged in competition, it is shown that competitive exclusion holds for any monotone growth response functions. Sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n-species case. In regard to the delay effects on the qualitative outcome of competition, it is demonstrated that when the delays are relatively small, the predictions of the model are identical with the predictions given by corresponding models without time delays. However, including large delays in the model may alter the predicted outcome of competition. The techniques used also work when different removal rates are permitted, an...