Abstract:
Overfishing can lead to the reduction or elimination of fish populations and the degradation or even destruction of their habitats. This can be prevented by introducing Marine Protected Areas (MPA's), regions in the ocean or along coastlines where fishing is controlled. MPA's can also lead to larger fish densities outside the protected area through spill-over, which in turn may increase the fishing yield. A natural question in this context, is where exactly to establish an MPA, in order to maximize these benefits. This problem is addressed along a one-dimensional stretch of coast-line, by first proposing a model for the fish dynamics. Fish are assumed to move diffusively, and are subject to recruitment, natural death and harvesting through fishing. The problem is then cast as an optimal control problem for the steady state equation corresponding to the PDE which models the fish dynamics. The functional being maximized is a weighted sum of the average fish density and the average fishing yield. It is shown that optimal controls exist, and that the form of an optimal control -and hence the location of the MPA- is determined by two key model parameters, namely the size of the coast, and the weight of the average fish density appearing in the functional. If these parameters are large enough -and precisely how large, can be calculated exactly- the results indicate when and where an MPA should be established. The results indicate that an MPA always takes the form of a Marine Reserve, where fishing is prohibited. The main mathematical tool used to prove the results is Pontryagin's maximum principle.

Abstract:
This paper investigates the effect of drug treatment on the standard within-host HIV model, assuming that therapy occurs periodically. It is shown that eradication is possible under these periodic regimes, and we quantitatively characterize successful drugs or drug combinations, both theoretically and numerically. We also consider certain optimization problems, motivated for instance, by the fact that eradication should be achieved at acceptable toxicity levels to the patient. It turns out that these optimization problems can be simplified considerably, and this makes calculations of the optima a fairly straightforward task. All our results will be illustrated by means of numerical examples based on up-to-date knowledge of parameter values in the model.

Abstract:
We provide a new global small-gain theorem for feedback interconnections of monotone input-output systems with multi-valued input-state characteristics. This extends a recent small-gain theorem of Angeli and Sontag for monotone systems with singleton-valued characteristics. We prove our theorem using Thieme's convergence theory for asymptotically autonomous systems. An illustrative example is also provided.

Abstract:
The tolerance of bacterial populations to biocidal or antibiotic treatment has been well documented in both biofilm and planktonic settings. However, there is still very little known about the mechanisms that produce this tolerance. Evidence that small, non-mutant subpopulations of bacteria are not affected by antibiotic challenge has been accumulating and provides an attractive explanation for the failure of typical dosing protocols. Although a dosing challenge can kill all the susceptible bacteria, the remaining persister cells can serve as a source of population regrowth. We give a robust condition for the failure of a periodic dosing protocol for a general chemostat model, which supports the mathematical conclusions and simulations of an earlier, more specialized batch model. Our condition implies that the treatment protocol fails globally, in the sense that a mixed bacterial population will ultimately persist above a level that is independent of the initial composition of the population. We also give a sufficient condition for treatment success, at least for initial population compositions near the steady state of interest, corresponding to bacterial washout. Finally, we investigate how the speed at which the bacteria are wiped out depends on the duration of administration of the antibiotic. We find that this dependence is not necessarily monotone, implying that optimal dosing does not necessarily correspond to continuous administration of the antibiotic. Thus, genuine periodic protocols can be more advantageous in treating a wide variety of bacterial infections.

Abstract:
We present a generalized Keller-Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller-Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.

Abstract:
We show that for certain configurations of two chemostats fed in parallel, the presence of two different species in each tank can improve the yield of the whole process, compared to the same configuration having the same species in each volume. This leads to a (so-called) "transgressive over-yielding" due to spatialization.

Abstract:
We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function.

Abstract:
Persistency is the property, for differential equations in $\R^n$, that solutions starting in the positive orthant do not approach the boundary. For chemical reactions and population models, this translates into the non-extinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory. Nontrivial examples are provided to illustrate the theory.

Abstract:
New checkable criteria for persistence of chemical reaction networks are proposed, which extend and complement those obtained by the authors in previous work. The new results allow the consideration of reaction rates which are time-varying, thus incorporating the effects of external signals, and also relax the assumption of existence of global conservation laws, thus allowing for inflows (production) and outflows (degradation). For time-invariant networks parameter-dependent conditions for persistence of certain classes of networks are provided. As an illustration, two networks arising in the systems biology literature are analyzed, namely a hypoxia and an apoptosis network.

Abstract:
We establish some properties of a within host mathematical model of malaria proposed by Recker et al which includes the role of the immune system during the infection. The model accounts for the antigenic variation exhibited by the malaria parasite (P. falciparum). We show that the model can exhibit a wide variety of dynamical behaviors. We provide criteria for global stability, competitive exclusion, and persistence. We also demonstrate that the disease equilibrium can be destabilized by non-symmetric cross-reactive responses.