Competition for resources is a fundamental topic in theoretical ecology. Population dynamics are coupled to dynamics of one or more resources by assuming a constant quota of nutrient per individual. In fact, quotas may vary, leading to variable-internal-stores models. When nutrient is taken up, it is stored internally, and population growth is a positive function of stored nutrient. The competitors live in a flowing habitat with both advection and diffusion, where the nutrient is supplied in the upstream flow, and all constituents flow out at the downstream end. The main difficulties in mathematical analysis for such systems are caused by the singularity at the extinction state. In this talk, we first investigate the nonlinear eigenvalue problem in the special positive cones of functions motivated by the ratio dependence. Then the threshold type result on the extinction/persistence of the species can be determined by the principal eigenvalue of our nonlinear eigenvalue problem. When the habitat is infinite, we shall attempt to study the travelling wave and spreading speeds.

 

References:

1.S.-B. Hsu, J. Jiang and F.-B. Wang, On a system of reaction- diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations 248 (2010), no. 10, 2470-2496.

2. S.-B. Hsu, K.-Y. Lam and F.-B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol. 75 (2017), no. 6-7, 1775-1825.

3. X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math. 60 (2007), pp. 1-40.