Global Dynamics of a Diffusive Two-Strain Epidemic Model with Non-Monotone Incidence Rate.

In this article, we investigate a diffusive two-strain epidemic model with non-monotone incidence rate and virus mutation. The positivity, existence and uniform boundedness of the solutions of the model system are studied. It is found that the system has three equilibrium points, namely the infection-free equilibrium point, the strain-2 endemic equilibrium point and both the strain-1 and strain-2 endemic equilibrium points. The global asymptotic stability analysis of the diffusive model system near all the equilibrium points is carried out by constructing appropriate Lyapunov functional. It is found that the system has no strain-1 endemic equilibrium point possibly due to the virus mutation. So, in this type of diseases, the infection due to strain-1 cannot be persistent in the community.

Balancing maximum sustainable yield and ecological resilience in an exploited two-predator one-prey system.

In this paper, we consider a two-predator one-prey system to determine the feedback of exploitation in individual as well as joint population levels. As balancing yield with resilience is highly essential for the conservation of species in the marine ecosystem, here we measure both the maximum sustainable yield (MSY) and resilience simultaneously. Then we investigate both the trade-offs and synergies among maximum yield, conservation, and resilience that emerge from different harvesting plans. It is found that for single species harvesting, a prey species-oriented system is capable of producing more yield in compare to any predator-oriented system but for resilience, a prey species-oriented system is far behind the others. In the case of joint harvesting of all the species, it is observed that the first predator-oriented system has a better ability to absorb the disturbances than the other cases. The correlation between yield and resilience at the MSY level is studied in all the cases. It is further observed that the increase of intraspecific competition in the predator decreases the risk of sustainability. In this way, this study may be helpful for fishery management to fulfill their goals without affecting the ecosystem's health in the long run.

Hydra effects in stable food chain models.

In this paper, we explore the occurrence of the hydra effect in food chains, a popular research theme in the current decade. The hydra effect, one of the paradoxical results in theoretical and applied ecology refers to the fact where increasing mortality rate on a population enhances its own stock. The main focus is to propose a dynamical system model of food chain showing a stable steady state and estimate the variation of stock of targeted species with increasing mortality. In our model, the per capita growth rate of any predator trophic level does not depend upon its density. The prey-predator model incorporating such a feature for predator growth is referred to as 'pure predator system' (see Sieber and Hilker (2012), J. Math. Biol. (2012) 64: 341-360, Journal of Mathematical Biology). Keeping the above feature in mind, we study a Rosenweig-MacArthur food chain model with logistic prey growth and Holling type II functional responses. It is shown that hydra effect at stable state appears on (a) prey in a four-trophic system, (b) first predator in a five-trophic system, and (c) prey and second predator in a six-trophic system. Xiao and Cao (2009) (Mathematical and Computer Modelling 50 (2009) 360-379) established that limit cycle may be observed due to harvesting in a system with the ratio-dependent prey-predator system (example of a non "non-pure predator system"). Therefore, if harvesting causes instability on some range of mortality rate, the hydra effect cannot occur at a stable state. Some results show that the unique stable steady state in our model remains stable under harvesting of either trophic level. As a whole, our investigations have some contribution in understanding population interactions, fishery management and biological pest control tactic.

Managing yield and resilience in a harvested tri-trophic food chain model.

In this article, we compare the two ecological services known as yield and resilience, for a tri-trophic food chain model consisting of a prey, an intermediate predator and a top predator. For this comparison process, we use both analytical and numerical techniques. It is shown that a variety of patterns are possible based on the intensity of efforts distributed among different trophic levels. Thus we may suggest that fishing down the food chain, as suggested by Pauly et al. (1998) is not bound to happen. Our analysis also shows that balancing the harvest between prey, intermediate predator and top predator could give more yield and stabilizing the ecosystem, than the selective harvesting of any one species. This balanced harvesting may not be a win-win situation for yield and resilience, but it could be a most favourable strategy to balance them. This research would help to correlate resilience with yield and determines the desirable selection of two policies, resilience maximizing yield or maximum sustainable yield to safeguard ecological communities.

Harvesting induced stability and instability in a tri-trophic food chain.

Non-equilibrium dynamics in the form of oscillations or chaos is often found to be a natural phenomenon in complex ecological systems. In this paper, we first analyze a tri-trophic food chain, which is an extension of the Rosenzweig-MacArthur di-trophic food chain. We then explore the impact of harvesting individual trophic levels to answer the following questions : a) when a non-equilibrium dynamics persists, b) whether it can locally be stabilized to a steady state, c) when the system switches from a stable steady state to a non-equilibrium dynamics and d) whether the Maximum Sustainable Yield (MSY) always exists when the top predator is harvested. It is shown that searching for a general theory to unify the harvesting induced stability must take into account the number of trophic levels and the degree of species enrichment, the outcomes that cannot be obtained from the earlier reports on prey-predator models. We also identify the situation where harvesting induces instability switching: the non-equilibrium state enters into a stable steady-state and then, upon more intensive harvesting, the steady-state again loses its stability. One of the new and important results is also that the MSY may not exist for harvesting the top predator. In general, our results contribute to biological conservation theory, fishery and ecosystem biodiversity management.

Biological conservation through marine protected areas in the presence of alternative stable states.

This article addresses how depleted stock can be restored by creation of marine reserve and species mobility when alternative stable states persist in a marine ecosystem. To understand the role of a marine protected area, we develop a two-patch version of an originally single-patch model. In the two-patch model, we prove that some of the locally stable equilibria are not stable equilibria from an ecological viewpoint. Similarly, some unstable equilibria determined classically from the mathematical model are no longer equilibria. It is shown that increasing reserve size may produce three alternative stable states in the presence of harvesting. Dynamic solutions have a tendency to reach an upper stable state from a lower stable state when reserve size is increased, but the opposite phenomenon (i.e., shifting to a lower stable state from an upper one) never occurs. This suggests that MPAs always have a positive effect in stock conservation even when alternative stable states inherently persist in marine ecosystems.