Lockdown is one of the drastic measures implemented by governments to curtail the spread of the Covid-19 pandemic and save lives. However, it has caused unprecedented damages to the economy. This paper provides a quantitative approach to assess the impact of a gradual, post-lockdown context concerning the spread of the disease. We propose to create a special class of individuals called “protected” who are risk-free to be infected. Such individuals could also be the vaccinated when an effective vaccine will be available. We developed a mathematical epidemic model for Covid-19 which describes the interactions between susceptible and infected individuals. We investigate the various and optimal strategies to curtail the spread of the infection at a minimum cost. As a case study on South Africa, the sensitivity analysis shows that investing on the special class “protected” is a better approach to reducing new secondary infections as opposed to reducing the contact rate between susceptible and infected individuals, or having more recovered patients. The simulations reveal that the peak could be reached in September 2020. This is consistent with the projection of the South African government as the winter season is expected to be over in mid August. Moreover, if 1 out of 1000 susceptible (cumulatively) join the special class, we project a maximum of 400,000 active cases. The number of infected and deaths could drastically increase as the proportion of individuals joining the special class decreases.

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This paper investigates the rich dynamics in a tritrophic food chain mathematical model, consisting of three species: prey, intermediate predators, and top predators. It is assumed that alternative food are supplied to intermediate predators in addition to feeding on prey. We consider a general Holling type response function and analyze the model. The existence and stability of six possible equilibrium points are established. These equilibrium points describe the various dynamics that could take place in the food chain. Hopf bifurcation, limit cycle, doubling periods, chaotic attractors, boundary crisis are observed in the numerical computations. Our results reveal the rich and complex dynamics of the interactions in the food chain. Recommendations for Resource ManagersOur investigation brings the followings to the attention of management:• Coexistence among predators and prey in the same environment is possible provided a good management of some factors (such as contacts between species, additional food supply, growth rate of species, etc).• The dynamics is complex and highly sensitive to the above factors. Strange or unpredictable behaviors could be observed.• The rate at which species are killed by a single predator (i.e., the functional responses) significantly affects the population sizes of all the species and the overall dynamics in the food chain. K E Y W O R D Scoexistence, food chain model, holling type functions, limit cycle, tritrophic

A model is proposed to understand the dynamics in a food chain (one predator‐two prey). Unlike many approaches, we consider mutualism (for defense against predators) between the two groups of prey. We investigate the conditions for coexistence and exclusion. Unlike Elettreby's (2009) results, we show that prey can coexist in the absence of predators (as expected since there is no competition between prey). We also show the existence of Hopf bifurcation and limit cycle in the model, and numerically present bifurcation diagrams in terms of mutualism and harvesting. When the harvest is practiced for profit making, we provide the threshold effort value ξ0 that determines the profitability of the harvest. We show that there is zero profit when the constant effort ξ0 is applied. Below (resp. above) ξ0, there will always be gain (resp. loss). In the case of gain, we provide the optimal effort ξ* and optimal steady states that produce maximum profit and ensure coexistence. Recommendations for resource managers As a result of our investigation, we bring the following to the attention of management: In the absence of predators, different groups of prey can coexist if they mutually help each other (no competition among them). There is a maximal effort ξ0 to invest in order to gain profit from the harvest. Above ξ0, the investment will result in a loss. In the case of profit from harvest, policy makers should recommend the optimal effort ξ* to be applied and the optimal stock (x1*,x2*,y*) to harvest. This will guarantee maximum profit while ensuring sustainability of all species.

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We analyse a predator-prey model with cubic growth in which animal movement are incorporated. We focus on the behaviour of the bands of animals to understand the global dynamic of the system. Travelling waves analyses are used to describe the time evolution of the system and to examine the interplay between the bands. We also highlight the influence of individuals behaviours on the collective behaviour of the bands. More importantly, we show the multi-dynamics that diffusion can cause, and we illustrate the patterns formed in the model as a result of a new phenomenon called transport-driven instability. This study shows how sustainable ecosystems could manipulate their movement characteristics to remain stable and viable.

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