In this work, we consider the heat equation coupled with Stokes equations under threshold type boundary condition. The conditions for existence and uniqueness of the weak solution are made clear. Next we formulate the finite element problem, recall the conditions of its solvability, and study its convergence by making use of Babuska–Brezzi's conditions for mixed problems. Third we formulate an Uzawa's type iterative algorithm that separates the fluid from heat conduction, study its feasibility, and convergence. Finally the theoretical findings are validated by numerical simulations.

Abstract:In this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of 'friction type'. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of BabuskaBrezzi's theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed varia-tional problem is presented and analyzed in details. Finally, numerical simulations that validate the theoret-ical findings are exhibited.

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.

This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.