Existence and uniqueness result for reaction-diffusion model of diffusive population dynamics

The present work investigates a convolution type nonlinear integrodifferential equation with diffusion. This type of equations represents not only pure mathematical interest, but also is widely used in various applications, especially in wide range of problems on population dynamics arising in biology. The existence of a parametric family of travelling wave solutions as well as the uniqueness of the solution in certain class of bounded continuous functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is proved. The study investigates also some important properties as well as asymptotic behaviour of constructed solutions. These results are then used to derive a new uniform estimate for the deviation between successive iterations, which provides us with a strong tool to control the number of iterations on our way of computing the desired numerical approximation of the exact solution. Finally, we apply our theoretical results to two well-known population problems modelled by delayed reaction-diffusion equation: Diffusion model for spatial-temporal spread of epidemics and stage structured population model.