Reaction-Diffusion Models with Allee Effects

Junping Shi (史峻平 )College of William and Mary,Williamsburg, Virginia, USAHarbin Normal University,

Harbin, HeLongJiang, China

(Joint work with R. Shivaji, Mississippi State University)

Workshop on Dynamical Systems and Bifurcation Theory, Shanghai Jiaotong University, Shanghai, China

May 25th, 2004

Problem we consider

A population whose density u(x,t) depends on the time t and the location x

Question: to persist, or extinct ?

Simplest Models

Thomas Robert Malthus

(1766-1834)

Malthus equation

Exponential growth

Pierre François Verhulst

(1804-1849)

Logistic growth

Bounded growth

Growth Rate Per Capita

If the growth equation is written as

Then f(P) is called growth rate per capita.

Growth rate per capita: (left) Malthus, (right) logistic

Allee effect

W.C.Allee [1938]: “…what minimal numbers are

necessary if a species is to maintain itself in nature?”

Growth rate is not always positive for small density,

and it may not be decreasing as in logistic model

either. The population is said to have an Allee effect,

if the growth rate per capita is initially an increasing

function, then decreases to zero at a higher density.

Strong and weak Allee effects

Weak Allee effect

f(0)>0

Strong Allee effect

f(0)<0

Dispersal of population: diffusion

Fisher [1937] Kolmogoroff-Petrovsky-Piscounoff [1937] Skellam [1951] Kierstead-Slobodkin [1953]

Reaction-diffusion Equation

Two phenomena of R-D equation

Wave propagation: on an unbounded habitat, the population will move from occupied area to unoccupied area with a constant velocity. This can be called as biological invasion, and eventually the entire territory is occupied by the species.

Critical patch size: on a bounded habitat with hostile boundary (u=0), the persistence of the population depends on the size of habitat. The population will persist if the size is larger than a critical number.

Diffusive logistic equation

Bifurcation Diagram for Diffusive Logistic Equation

A more general approach

Bifurcation Diagram in general

Bounds of bifurcation curve

The bifurcation curve is bounded by two monotone curves, which are curves for logistic growth equations as in the following diagrams,

(a) Weak Allee effect; (b) upper logistic; (c) lower logistic.

Exact Multiplicity of Solutions

Ouyang-Shi [1998, 1999] JDE

Korman-Shi [2000] Proc. Royal Soc. Edin.

Transition from extinction to persistence Hysteresis occurs when lambda decreases.

Ludwig-Aronson-Weinberger

[1978]

Spruce Budworm model

Example 1

Diffusive logistic equation with predation functional response of type II

f(u) can be logistic, weak or strong Allee effect

Ball: Korman-Shi [2000] General habitat: Shi-Shivaji [2004]

Example 2

K(x) zero: many many works [Ouyang, 1992]K(x) not zero: [Alama-Tarantello, 1995]

Example 3: Nonlinear Diffusion Equation

Aggregation-induced weak Allee effect

Exact Multiplicity for Nonlinear Diffusion Equation

NSF Undergraduate Research Project (Spring 2004)

Young-He Lee (Class of 2004), Lena Sherbakov (Class of 2005),

Jacky Taber (Class of 2006), Junping Shi

Exact Multiplicity of solution can be studied via a generalized time-mapping

When D(u)=u^2-u+1 and f(u) is weak Allee effect, there are four solutions for certain lambda