Oscillations of two competing microbial populations in configurations of two interconnected chemostats

CHAOTIC COEXISTENCE IN A RESOURCE–CONSUMER MODEL

We study the interspecies competition in a simple resource–consumer model which includes the resource supply as a dynamic variable. In the model, an organism of each species needs to consume a certain minimum rate of resource (food) to survive; a higher rate of consumption is required for reproduction. We analyze the orbit diagrams and Lyapunov exponents in order to characterize the system dynamical behavior. We observe that under a periodic food supply, the system can exhibit coexistence in the form of limit cycle oscillations. For a certain parameter range, we observe chaotic behavior emerging from successive period doublings and quasi-periodicity.

Dynamics of the stochastic chemostat with Monod-Haldane response function

The stochastic chemostat model with Monod-Haldane response function is perturbed by environmental white noise. This model has a global positive solution. We demonstrate that there is a stationary distribution of the stochastic model and the system is ergodic under appropriate conditions, on the basis of Khasminskii’s theory on ergodicity. Sufficient criteria for extinction of the microbial population in the stochastic system are established. These conditions depend strongly on the Brownian motion. We find that even small scale white noise can promote the survival of microorganism populations, while large scale noise can lead to extinction. Numerical simulations are carried out to illustrate our theoretical results.

Global dynamics of the buffered chemostat for a general class of response functions

We study how a particular spatial structure with a buffer impacts the number of equilibria and their stability in the chemostat model. We show that the occurrence of a buffer can allow a species to persist or on the opposite to go extinct, depending on the characteristics of the buffer. For non-monotonic response functions, we characterize the buffered configurations that make the chemostat dynamics globally asymptotically stable, while this is not possible with single, serial or parallel vessels of the same total volume and input flow. These results are illustrated with the Haldane kinetic function.

Complex dynamics of microbial competition in the gradostat

We examine the conditions necessary for the emergence of complex dynamic behavior in systems of microbial competition. In particular, we study the effect of spatial heterogeneity and substrate-inhibition on the dynamics of such a system. This is accomplished through the study of a mathematical model of two microbial populations competing for a single nutrient in a configuration of two interconnected chemostats. Microbial growth is assumed to follow substrate-inhibited kinetics for both species. Such a system with sterile feed has been shown in a previous work to exhibit stable periodic states. In the present work we study the system for the case of non-sterile feed, i.e., when the two species are present in the feed of the chemostats. The analysis is done by numerical bifurcation theory methods. We demonstrate that, in addition to periodic states, the system possesses stable quasi-periodic states resulting from Neimark-Sacker bifurcations of limit cycles. Also, periodic states may undergo successive period doublings leading to periodic states of increasing period and indicating that chaotic states might be possible. Multistability is also observed, consisting in the coexistence of several stable steady states and possibly stable periodic or quasi-periodic states for given operating conditions. It appears that substrate-inhibition, spatial heterogeneity and presence of microorganisms in the inflow are all necessary conditions for complex dynamics to arise in a microbial system of pure and simple competition.

Dispersal-mediated coexistence of competing predators

Models of metapopulations have often ignored local community dynamics and spatial heterogeneity among patches. However, persistence of a community as a whole depends both on the local interactions and the rates of dispersal between patches. We study a mathematical model of a metacommunity with two consumers exploiting a resource in a habitat of two different patches. They are the exploitative competitors or the competing predators indirectly competing through depletion of the shared resource. We show that they can potentially coexist, even if one species is sufficiently inferior to be driven extinct in both patches in isolation, when these patches are connected through diffusive dispersal. Thus, dispersal can mediate coexistence of competitors, even if both patches are local sinks for one species because of the interactions with the other species. The spatial asynchrony and the competition-colonization trade-off are usual mechanisms to facilitate regional coexistence. However, in our case, two consumers can coexist either in synchronous oscillation between patches or in equilibrium. The higher dispersal rate of the superior prompts rather than suppresses the inferior. Since differences in the carrying capacity between two patches generate flows from the more productive patch to the less productive, loss of the superior by emigration relaxes competition in the former, and depletion of the resource by subsidized consumers decouples the local community in the latter.

Coexistence of three microbial populations competing for three complementary nutrients in a chemostat

We study a model of three microbial populations competing for three complementary nutrients in a single chemostat. By using methods of numerical bifurcation theory we analyze the model equations and determine the effect of the model parameters on the dynamics of the system. The main question to be answered is whether there exist conditions under which the three populations can coexist in a stable state in the chemostat. The analysis shows that coexistence can be obtained as a stable steady state but also as a stable periodic state for a wide range of operating conditions of the chemostat.

Computing operating diagrams of bioreactors

The operating diagram of a bioreactor is an illustrative way to present the effect that the operating conditions have on its long-term behavior. It can be constructed if a mathematical model of the bioreactor is available. The procedure for constructing the operating diagram consists in analyzing the dynamic behavior of the system of the differential equations of the model. Some methods are described that can be used in computing operating diagrams of bioreactors. They are based on numerical bifurcation techniques for systems of differential equations. Both cases of bioreactors with constant and periodically varying operating conditions are considered.

On the coexistence of three microbial populations competing for two complementary substrates in configurations of interconnected chemostats

We examine the question of coexistence of three microbial populations competing for two complementary rate-limiting substrates in configurations of interconnected chemostats. It is known that coexistence of two populations competing for two rate-limiting substrates is possible in a single chemostat, but coexistence is not possible when three populations are involved. We examine whether coexistence of three populations becomes possible by considering configurations of two or three interconnected chemostats, thus allowing for effects of spatial heterogeneity. Computational analysis of the model equations indicates that in the case of two chemostats coexistence is possible only for specific discrete parameter values where the system is structurally unstable and therefore the coexistence state is not practically attainable, whereas in the case of three chemostats coexistence is possible for a whole range of parameter values where the system is structurally stable and therefore the coexistence state can be realized in practice.