Biology is at the centre of the current advancement of all natural sciences. Many branches of biology, such as cellular and molecular biology, evolution and genetics, developmental biology, and ecology and conservation, require qualitative and quantitative analysis beyond standard tools from classical mathematics, physics and chemistry. Mathematical biology is an interdisciplinary research direction for developing new mathematical and computational methods, theory to answer questions arisen from modern biological and biomedical research.
Mathematical modelling of biological phenomena can be traced to the population models of Malthus, Verhulst, Lokta and Volterra, genetic laws of Hardy, Weinberg, Fisher, Haldane and Wright, epidemic models of Ross, McKendrick and Kermack, genetic spatial dispersal models of Fisher, Kolmogorov, Petrovskii, and Piscounov, action potentials in neuron model of Hodgkin and Huxley, and morphogenesis pattern formation theory of Turing. New computing power has made mathematical biology or biomathematics an important scientific branch with tremendous growth in recent years, and the modelling and analysis effort have mixed mathematical theory and techniques, physical/chemical experiments and computer simulations into a truly interdisciplinary research.
This special issue of ‘Analysis and Applications of Partial Differential Equations in Biomathematics’ in the journal Applicable Analysis collects a variety of works which showcase the current state of art of mathematical biology analysis, especially the models of partial differential equations and variants which model ecological, physiological and biochemical phenomena. Here we briefly summarize the collection according to their topics.
The competitive model is the one in which several competitors compete for one or more common resource, and the well-known Lotka–Volterra equation of competition has been used for many applications, such as chemostatic bio-reactor, and phytoplankton competition. The reaction-diffusion competitive models incorporate the random dispersal of the organisms in the environment, hence they are more realistic representation of the real competition phenomena. Three contributions in the special issue consider reaction-diffusion competitive models, but each with a distinctive new angle. Bezuglyy and Lou study a competitive model with diffusion and advection, as well as the corresponding single species model. Hence in addition to the random movement described by diffusion, biased movement of species upward along the resource gradient is modelled with advection. A competition exclusion principle is proved for two species which are identical except for their dispersal strategies. Kuto and Yamada also consider Lotka–Volterra type competition model with not only diffusion but also the cross-diffusion. They prove that when the cross-diffusion coefficients tend to infinity, then positive solutions of the competition system converge to a positive solution of a suitable limiting system. Nie and Wu consider a reaction-diffusion chemostat model with two competing organisms competing for the same nutrient. They use Lyapunov–Schmidt procedure and perturbation technique to prove the uniqueness and stability of the coexistence steady state solution for the unstirred chemostat model.
Besides the advection and cross-diffusion variations of the reaction-diffusion models mentioned above, non-local reaction-diffusion models have been used to model not only the short range random dispersal but also the long range dispersal. Ai's work considers the existence of periodic and homoclinic solutions of a general class of non-local reaction-diffusion models using perturbation methods, and these solutions are shown to be persistent from the unperturbed equation (which is classical reaction-diffusion equation). Lutscher's work considers non-local integro-differential equations for population spread and persistence in heterogeneous landscapes, and he develops appropriate averaging methods for averaging over landscape and patch scales.
Mathematic models of epidemics spreading are of current importance with the continuous looming of influenza, HIV, ebola, dengue fever, etc. Lou and Zhao reconsider the classical Ross–MacDonald model of malaria transmission, and they study a model with diffusion of mosquitoes and human and time-periodic environment. They prove results for the spreading speed of the travelling wave solutions, and also threshold dynamics for the bounded spatial domain. Magal, McCluskey and Webb propose an infection-age model of disease transmission, where both the infectiousness and the removal rate may depend on the infection-age. They use the Lyapunov functional technique to show the global stability of the disease-free and endemic equilibrium points when the basic reproduction number is less than or greater than 1.
Su, Wan and Wei consider the Hopf bifurcation of periodic orbits of a reaction-diffusion model with delay effect in a food-limited population for which the growth limitations are based upon the proportion of available resources not utilized. Feng and Wan consider some generalized reaction-diffusion equations with higher-order nonlinear rate of growth, and they use generalized linearizing transformation to derive first integrals of the system and consequent explicit travelling wave solutions. Kurata and Matsuzawa study the steady state equation of a reaction-diffusion equation with a balanced bistable nonlinearity and heterogeneous environment, and they show the existence of non-trivial stable solution with rich spatial pattern.
We feel that the mathematical techniques demonstrated in these articles are good representations of the current development in partial differential equations, dynamical systems, bifurcation theory, eigenvalue problems, perturbation methods, and they are effectively applied to a variety of biological models with organism dispersal playing an essential role. This clearly shows the pivotal importance of the advanced mathematical methods in the rapidly developing life sciences.