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Reaction-Diffusion Models with Allee Effects

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)

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Reaction-Diffusion Models with Allee Effects

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  1. 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) Workshop on Dynamical Systems and Bifurcation Theory, Shanghai Jiaotong University, Shanghai, China May 25th, 2004

  2. Problem we consider • A population whose density u(x,t) depends on the time t and the location x • Question: to persist, or extinct ?

  3. Simplest Models Thomas Robert Malthus (1766-1834) Malthus equation Exponential growth Pierre François Verhulst (1804-1849) Logistic growth Bounded growth

  4. 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

  5. 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.

  6. Strong and weak Allee effects Strong Allee effect f(0)<0 Weak Allee effect f(0)>0

  7. Dispersal of population: diffusion • Fisher [1937] • Kolmogoroff-Petrovsky-Piscounoff [1937] • Skellam [1951] • Kierstead-Slobodkin [1953] Reaction-diffusion Equation

  8. 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.

  9. Diffusive logistic equation

  10. Bifurcation Diagram for Diffusive Logistic Equation

  11. A more general approach

  12. Bifurcation Diagram in general

  13. 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.

  14. Exact Multiplicity of Solutions Ouyang-Shi [1998, 1999] JDE Korman-Shi [2000] Proc. Royal Soc. Edin.

  15. Transition from extinction to persistence Hysteresis occurs when lambda decreases. Ludwig-Aronson-Weinberger [1978] SpruceBudworm model

  16. 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]

  17. Example 2 K(x) zero: many many works [Ouyang, 1992] K(x) not zero: [Alama-Tarantello, 1995]

  18. Example 3: Nonlinear Diffusion Equation Aggregation-induced weak Allee effect

  19. 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

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