Narrow Results

In this paper, we introduce a partial differential equation (PDE) model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans. The model features the variable infectiousness during the infectious period, which we call the infection age of the infectious host. We define two thresholds ${\mathcal{ℛ}}_1^j$ and ${\mathcal{ℛ}}_2^j,j=1,2$, and show that the strain $j$ can not invade the system if ${\mathcal{ℛ}}_1^j+{\mathcal{ℛ}}_2^j<1$. Further, the disease-free equilibrium of the system is globally asymptotically stable if ${\mathrm{\text{max}}}_j\{{\mathcal{ℛ}}_1^j+{\mathcal{ℛ}}_2^j\}<1$. When ${\mathcal{ℛ}}_1^j>1$, strain $j$ dominance equilibrium ${\mathcal{ℰ}}_j$ exists, and is locally asymptotically stable when ${\mathcal{ℛ}}_1^j>1$, ${\mathcal{ℛ}}_1^i<\varsigma {\mathcal{ℛ}}_1^j,i,j=1,2,i\ne j$, $\varsigma \in (0,1)$. Then, by applying Lyapunov–LaSalle techniques, we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain $j$. This implies strain $j$ eliminates the other strain as long as ${\mathcal{ℛ}}_1^i/{\mathcal{ℛ}}_1^j<b_i/b_j<1,i\ne j$, where $b_j$ denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain $j$. Finally, we study the existence of the coexistence equilibria under some conditions.

Toxic or allelopathic compounds liberated by toxin-producing phytoplankton (TPP) acts as a strong mediator in plankton dynamics. On an analysis of a set of phytoplankton biomass data that have been collected by our group in the northwest part of the Bay of Bengal, and by analysis of a three-component mathematical model under a constant as well as a stochastic environment, we explore the role of toxin-allelopathy in determining the dynamic behavior of the competing phytoplankton species. The overall results, based on analytical and numerical wings, demonstrate that toxin-allelopathy due to the TPP promotes a stable co-existence of those competitive phytoplankton that would otherwise exhibit competitive exclusion of the weak species. Our study suggests that TPP might be a potential candidate for maintaining the co-existence and diversity of competing phytoplankton species.

We consider a chemostat model of phytoplankton competing for nitrogen taking into account effects of both intra- and interspecific crowding and the light limitation. We consider crowding as an additive density-dependent mortality rate. Crowding effects may be classified into intra- and interspecific crowding depending on whether the additional mortality is caused by the same or alternate species.

We analyze the existence and local and global stability of single species and coexistence equilibria using the linearization and stability method of Lyapunov. We present a numerical example illustrating the fact that the crowding effects may lead to the bistable coexistence of two phytoplankton species. We demonstrate that the crowding effects and the light limitation affect the outcome of exploitative competition for a single resource and promote coexistence. We also show that while the crowding has a stabilizing effect on phytoplankton community, the light limitation may destabilize the system and produce sustained oscillations.

In this paper we consider the problem of reconstructing separatrices in dynamical systems. In particular, here we aim at partitioning the domain approximating the boundaries of the basins of attraction of different stable equilibria. We start from the 2D case sketched in Cavoretto et al. [2011] and the approximation scheme presented in [Cavoretto et al. (2011); Cavoretto et al. (2015)], and then we extend the reconstruction scheme of separatrices in the cases of three-dimensional models with two and three stable equilibria. For this purpose we construct computational algorithms and procedures for the detection and the refinement of points located on the separatrix manifolds that partition the phase space. The use of the so-called Radial Basis Function (RBF) Partition of Unity (PU) method is used to reconstruct the separatrices.