%0 Journal Article %1 cantrell1991diffusive %A Cantrell, Robert Stephen %A Cosner, Chris %D 1991 %J SIAM J. Math. Anal. %K diffusive_logistic_equation eigenvalue_problem fishers_diffusion probability_of_survival spatial_structure %N 4 %P 1043--1064 %R 10.1137/0522068 %T Diffusive logistic equations with indefinite weights: population models in disrupted environments. II %U http://dx.doi.org/10.1137/0522068 %V 22 %X The dynamics of a population inhabiting a strongly heterogeneous environment are modeled by diffusive logistic equations of the form $u_1 = (d(x,u) + u) - b(x) u + m(x)u - cu^2 $ in $Ømega (0,)$, where u represents the population density, $d(x,u)$ the (possibly) density dependent diffusion rate, $b(x)$ drift, c the limiting effects of crowding, and $m(x)$ the local growth rate of the population. The growth rate $m(x)$ is positive on favorable habitats and negative on unfavorable ones. The environment $Ømega $ is bounded and surrounded by uninhabitable regions, so that $u = 0$ on $Ømega (0,)$. In a previous paper, the authors considered the special case $d(x,u) = d$, a constant, and $b = 0$, and were able to make an analysis based on variational methods. The inclusion of density dependent diffusion and/or drift makes for more flexible and realistic models. However, variational methods are mathematically insufficient in these more complicated situations. By employing methods based on monotonicity and positive operator theory, many previous results on the dependence on m of the overall suitability of the environment can be recovered and some new results can be established concerning environmental quality dependence on $b$. In the process, a bifurcation and stability analysis is made of the model which includes some new estimates on eigenvalues for associated linear problems.

@article{cantrell1991diffusive, abstract = {The dynamics of a population inhabiting a strongly heterogeneous environment are modeled by diffusive logistic equations of the form $u_1 = \nabla \cdot (d(x,u) + \nabla u) - {\bf b}(x) \cdot \nabla u + m(x)u - cu^2 $ in $\Omega \times (0,\infty )$, where u represents the population density, $d(x,u)$ the (possibly) density dependent diffusion rate, ${\bf b(x)}$ drift, c the limiting effects of crowding, and $m(x)$ the local growth rate of the population. The growth rate $m(x)$ is positive on favorable habitats and negative on unfavorable ones. The environment $\Omega $ is bounded and surrounded by uninhabitable regions, so that $u = 0$ on $\partial \Omega \times (0,\infty )$. In a previous paper, the authors considered the special case $d(x,u) = d$, a constant, and ${\bf b} = 0$, and were able to make an analysis based on variational methods. The inclusion of density dependent diffusion and/or drift makes for more flexible and realistic models. However, variational methods are mathematically insufficient in these more complicated situations. By employing methods based on monotonicity and positive operator theory, many previous results on the dependence on m of the overall suitability of the environment can be recovered and some new results can be established concerning environmental quality dependence on ${\bf b}$. In the process, a bifurcation and stability analysis is made of the model which includes some new estimates on eigenvalues for associated linear problems.}, added-at = {2013-03-13T13:09:56.000+0100}, author = {Cantrell, Robert Stephen and Cosner, Chris}, biburl = {https://www.bibsonomy.org/bibtex/2467a63a5f0b9d440b92741de61e3fd0e/peter.ralph}, coden = {SJMAAH}, doi = {10.1137/0522068}, fjournal = {SIAM Journal on Mathematical Analysis}, interhash = {4e2dd1d31cb468fffa7d638fc4868f03}, intrahash = {467a63a5f0b9d440b92741de61e3fd0e}, issn = {0036-1410}, journal = {SIAM J. Math. Anal.}, keywords = {diffusive_logistic_equation eigenvalue_problem fishers_diffusion probability_of_survival spatial_structure}, mrclass = {92D40 (35B32 35B35 35K60 92D25)}, mrnumber = {1112065 (92h:92015)}, number = 4, pages = {1043--1064}, timestamp = {2014-07-01T02:42:27.000+0200}, title = {Diffusive logistic equations with indefinite weights: population models in disrupted environments. {II}}, url = {http://dx.doi.org/10.1137/0522068}, volume = 22, year = 1991 }