Differential Equations, Dynamical Systems, and Mathematical Biology
More precisely: Qualitative studies of nonlinear partial differential equations -- mainly parabolic equations -- that arise in biology, with emphasis on the global and dynamical structure of solutions.
- Liang Kong and Wenxian Shen, Positive Stationary Solutions and Spreading Speeds of KPP Equations in Locally Spatially Inhomogeneous Media, Methods and Applications of Analysis, Vol. 18, No. 4 (2011), pp. 427-456. (PDF).
This paper mainly explores spatial spread and front propagation dynamics of KPP evolution equations with random or nonlocal or discrete dispersal in unbounded inhomogeneous and random media and reveals such an important biological scenario: the localized spatial in-homogeneity of the media does not prevent the population to persist and to spread, moreover, it neither slows down nor speeds up the spatial spread of the population. Dispersal of organisms is one of key component of ecological and evolutionary processes. Hence this paper is concerned with 3 types of dispersals: random, nonlocal and discrete (in the patchy environment).
- Liang Kong and Wenxian Shen, Time Periodic Positive Solutions and Spreading Speeds of KPP Equations in Temporally periodic and Locally Spatially Inhomogeneous Media, Preprint.
This paper mainly considers spatial spread and front propagation dynamics of KPP evolution equations with random or nonlocal or discrete dispersal in unbounded inhomogeneous and random media, which involving both temporal and spatial heterogeneity.
- (Joint with Wenxian Shen) Analysis of spreading speed for competition models in some spatially heterogeneous environment, in preparation.
It is well known about the spatial spread of organisms in a single species case, however very little is known about how species interactions influence the spatial spread of invasive species, especially in the heterogeneous (say, spatially locally inhomogeneous) environments. Authors derive a set of sufficient conditions for linear determinacy in spatially locally inhomogeneous two-species competition models. When these conditions are not satisfied, spread rate may exceed linearly determined predictions.
- A. Bolotskikh, L. Kong, R. Mess, F. Bill Shi, Y. Zhang, Models for Predicting Indoor Pollution from Questionnaire and Outdoor Pollution Data: The Inner-City Asthma Air Pollution Study. Proceedings of the Industrial Mathematical and Statistical Modeling Workshop, SAMSI (2011)(PDF).