| Summary: | This thesis consists of two parts. In the first part of the thesis, we study a local-nonlocal Fisher-Komogorov-Petrovskii-Piskunov equation, which contains a linear combination of local and nonlocal interaction terms in one spatial dimension. Similar to the local case, this equation admits two uniform steady states, one stable and the other unstable. In our study, we restrict our attention to kernels that are symmetric, strictly positive, and decreasing with distance to describe the nonlocal interaction over space. Using asymptotic and numerical analysis, we show that travelling wave solutions of this model have interesting properties when the diffusion length scale is much smaller than the interaction length scale, and the model is highly nonlocal. For example, we observe non-monotonic travelling waves behind the wavefront, connecting the two steady states. In particular, the solutions behave like Dirac delta functions where the height tends to infinity and the width tends to zero in the same limit before converging to the stable steady state.
In the second part of the thesis, we analyse the stability of the spatially-uniform, coexisting steady state of a diffusive nonlocal Lotka-Volterra system for two species in one spatial dimension, and include terms that model both nonlocal intraspecific and local interspecific competition. We find that the coexisting state of the system can lose stability once nonlocality is introduced. We use asymptotic and numerical analysis to find the neutral curve and also derive amplitude equations using weakly nonlinear analysis. These are used to determine bifurcation behaviour close to the neutral curve and confirmed to be consistent with behaviour observed in direct numerical simulations. Finally, we construct the asymptotic solution in the limit of weak diffusivity and find that the leading order periodic solution consists of disjoint regions where either one or the other species is absent.
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