关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

Many ``real-world'' networks are clearly defined while most ``social'' networks are to some extent subjective. Indeed, the accuracy of empirically-determined social networks is a question of some concern because individuals may have distinct perceptions of what constitutes a social link. One unambiguous type of connection is sexual contact. Here we analyze data on the sexual behavior of a random sample of individuals, and find that the cumulative distributions of the number of sexual partners during the twelve months prior to the survey decays as a power law with similar exponents $\alpha \approx 2.4$ for females and males. The scale-free nature of the web of human sexual contacts suggests that strategic interventions aimed at preventing the spread of sexually-transmitted diseases may be the most efficient approach.

The Web of Human Sexual Contacts

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Diffusion, Self-Diffusion and Cross-Diffusion

To understand the impact of spatial heterogeneity of environment and movement
 of individuals on the persistence and extinction of a disease, a spatial SIS
 reaction-diffusion model is studied, with the focus on the existence,
 uniqueness and particularly the asymptotic profile of the steady-states. First,
 the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is
 shown that if $\R_{0} < 1$, the unique disease-free equilibrium is globally
 asymptotic stable and there is no endemic equilibrium. If $\R_{0} > 1$, the
 disease-free equilibrium is unstable and there is a unique endemic equilibrium.
 A domain is called high (low) risk if the average of the transmission rates is
 greater (less) than the average of the recovery rates. It is shown that the
 disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk
 domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0}
 < 1)$ if and only if infected individuals have mobility above a threshold
 value. The endemic equilibrium tends to a spatially inhomogeneous disease-free
 equilibrium as the mobility of susceptible individuals tends to zero.
 Surprisingly, the density of susceptibles for this limiting disease-free
 equilibrium, which is always positive on the subdomain where the transmission
 rate is less than the recovery rate, must also be positive at some (but not
 all) places where the transmission rates exceed the recovery rates.

Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model

Diffusion vs cross-diffusion : an elliptic approach

On the effects of migration and spatial heterogeneity on single and multiple species

Random dispersal is essentially a local
behavior which describes the movement of organisms between
adjacent spatial locations. However, the movements and
interactions of some organisms can occur between non-adjacent
spatial locations. To address the question about which dispersal
strategy can convey some competitive advantage, we consider a
mathematical model consisting of one reaction-diffusion equation
and one integro-differential equation, in which two competing
species have the same population dynamics but different dispersal
strategies: the movement of one species is purely by random walk
while the other species adopts a non-local dispersal strategy.
For spatially periodic and heterogeneous environments we show
that (i) for fixed random dispersal rate,
if the nonlocal dispersal distance is sufficiently small, then
the non-local disperser can invade the random disperser but not vice versa;
(ii) for fixed nonlocal dispersal distance,
if the random dispersal rate becomes sufficiently small,
 then
the random disperser can invade the nonlocal disperser but not
vice versa.
These results suggest
that for spatially periodic heterogeneous environments,
the competitive advantage may belong to the species with much lower
effective rate of dispersal. This is in agreement with previous results
for the evolution of random dispersal [9, 13] that the slower disperser
 has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile
surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.

/pdf/random-dispersal-vs-non-local-dispersal-2315myzfyw.pdf

Yuan Lou

Papers

Diffusion, Self-Diffusion and Cross-Diffusion

Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model

Diffusion vs cross-diffusion : an elliptic approach

On the effects of migration and spatial heterogeneity on single and multiple species

Random dispersal vs. non-local dispersal