reaction-diffusion models with allee effects
DESCRIPTION
Reaction-Diffusion Models with Allee Effects. Junping Shi ( 史峻平 ) College of William and Mary, Williamsburg, Virginia, USA Harbin Normal University, Harbin, HeLongJiang, China (Joint work with R. Shivaji, Mississippi State University) - PowerPoint PPT PresentationTRANSCRIPT
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