A diffusive logistic equation with U-shaped density dependent dispersal on the boundary

Jerome Goddard II, Quinn Morris, Catherine Payne, Ratnasingham Shivaji

DOI: http://dx.doi.org/10.12775/TMNA.2018.047


We study positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} - \Delta v = \lambda v(1-v), & x \in \Omega_0, \\ \displaystyle \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} ( v-A)^2 v =0 , & x \in \partial \Omega_0, \end{cases} \end{equation*} where $\Omega_0$ is a bounded domain in $\mathbb{R}^n$; $n \ge 1$ with smooth boundary $\partial \Omega_0$, ${\partial }/{\partial \eta}$ is the outward normal derivative, $A \in (0,1)$ is a constant, and $\lambda$, $\gamma$ are positive parameters. Such models arise in the study of population dynamics when the population exhibits a U-shaped density dependent dispersal on the boundary of the habitat. We establish existence, multiplicity, and uniqueness results for certain ranges of the parameters $\lambda$ and $\gamma$. We obtain our existence and mulitplicity results via the method of sub-super solutions.


Mathematical biology; reaction diffusion model; nonlinear boundary conditions; U-shaped density dependent dispersal

Full Text:



R.S. Cantrell and Ch. Cosner, On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differential Equations 231 (2006), 768–804.

R.S. Cantrell and Ch. Cosner, Density dependent behavior at habitat boundaries and the Allee effect, Bull. Math. Biol. 69 (2007), 2339–2360.

J.T. Cronin, Movement and spatial population structure of a prairie planthopper, Ecology 84 (2003), no. 5, 1179–1188.

J.T. Cronin, J. Goddard II and R. Shivaji, Effects of patch matrix and individual movement response on population persistence at the patch-level (2017), preprint.

K. Enfjall and O. Leimar, Density-dependent dispersal in the Glanville fritillary, Melitaea cinxia, Oikos 108 (2005), no. 3, 465–472.

J. Goddard II, K. Ashley and V. Sincavage, Ecological systems, nonlinear boundary conditions, and sigma-shaped bifurcation curves, Involve 6 (2013), no. 4, 399–430.

J. Goddard II, E.K. Lee and R. Shivaji, Diffusive logistic equation with non-linear boundary conditions, J. Math. Anal. Appl. 375 (2011), 365–370.

J. Goddard II, E.K. Lee and R. Shivaji, Population models with nonlinear boundary conditions, Electron. J. Differential Equations, Conference 19 (2010), 135–149.

J. Goddard II, E.K. Lee and R. Shivaji, Population models with diffusion, strong Allee effect, and nonlinear boundary conditions, Nonlinear Anal. 74 (2011), no. 17, 6202–6208.

J. Goddard II, Q. Morris, S. Robinson and R. and Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics (2017). (submitted)

J. Goddard II, J. Price and R. Shivaji, Analysis of steady states for classes of reactiondiffusion equations with U-shaped density dependent dispersal on the boundary, (2017), preprint.

J. Goddard II and R. Shivaji, Halo-shaped bifurcation curves in ecological systems, Electron. J. Differential Equations 2014 (2014), no. 88, 1–27.

J. Goddard II and R. Shivaji, Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary, J. Math. Anal. Appl. 414 (2014), no. 2, 561–573.

F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J. 31 (1982), no. 2, 213–221.

S.-Y. Kim, R.Torres and H. Drummond, Simultaneous positive and negative densitydependent dispersal in a colonial bird species, Ecology 90 (2009), no. 1, 230–239.

M. Kuussaari, M. Nieminen and I. Hanski, An experimental study of migration in the Glanville Fritillary butterfly Melitaea cinxia, J. Anim. Ecol. 65 (1996), no. 6, 791–801.

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville Fritillary butterfly, Oikos 82 (1998), no. 2, 384–392.

D. Ludwig, D.G. Aronson and H.F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol. 8 (1979), no. 3, 217–258.

G. Maciel and F. Lutscher, How individual movement response to habitat edges affects population persistence and spatial spread Amer. Nat. 182 (2013), no. 1, 42–52

F.J. Odendaal, P. Turchin and F.R. Stermitz, Influence of host-plant density and male harassment on the distribution of female Euphydryas anicia (Nymphalidae), Oecologia 78 (1989), no. 2, 283–288.

M.A. Rivas and S. Robinson, Eigencurves for linear elliptic equations European Journal ESAIM, (to appear).

A.M. Shapiro, The role of sexual behavior in density-related dispersal of Pierid butterflies, American Naturalist 104 (1970), no. 938, 367–372.


  • There are currently no refbacks.

Partnerzy platformy czasopism