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ROCKY MOUNTAIN MATHEMATICS CONSORTIUM SUMMER SCHOOL 2012 MATHEMATICAL MODELING IN ECOLOGY AND EPIDEMIOLOGY UNIVERSITY OF WYOMING JUNE 11-22, 2012. II. GENERAL THEORY OF AGE-STRUCTURED POPULATION DYNAMICS.
SUMMER SCHOOL 2012
MATHEMATICAL MODELING IN ECOLOGY AND EPIDEMIOLOGY
UNIVERSITY OF WYOMING
JUNE 11-22, 2012
II. GENERAL THEORY OF AGE-STRUCTURED POPULATION DYNAMICS
1. The classical logistic equation describes the dynamics of a population whose inherent growth rate is constrained by increased mortality as the population increases above the environmental carrying capacity.
P(t) = population at time t
l1=inherent (Malthusian) growth rate
l1 C = environmental carrying capacity
P0 = population at time 0
The solution of the logistic equation for a bacteria population. l1= ln 2 /.5 (the doubling time is 30 min = .5 hours) C = 1011 = the carrying capacity = l1C = 1.3869 x 1011P0 = 10 (initially there are 10 bacteria)
u’(t) = Au(t) -Fu(t) u(t), t> 0, u(0) = x in X
where X is a Banach lattice, A is the infinitesimal generator of a strongly
continuous semigroup of bounded linear operators in X, and F is in X*+
(X+ is the positive cone of X, X*is the dual of X, and X*+ is the positive
cone of X*).
Definition. A semigroup of positive linear (nonlinear) operators in X is
a family T(t), t >0 of positive linear (nonlinear) operators in X satifying
(1) T(0)x = x for x in X
(2) T(t+s) x = T(t) T(s)x for s, t >0
(3) T(t)x is continuous from R+ to X for each x in X.
The formula for u(t) is defined for all x in X+ and S(t)x = u(t) defines a semigroup of postive nonlinear operators S(t), t> 0 in X+ .
X = R, Ax = l1 x,Fx = x/C, T(t)x = exp(l1 t) x, and
Observe that if P1 has 1-dimensiona1 range, then there exists x1 in X such that P1x = cx1 for all x in X, where c is a constant depending on x. If we choose x1 in X such that Fx1= 1, then
is independent of x in X.
If P1 has 1-dimensiona1 range, then the linear semigroup T(t), t > 0 is said to have asynchronous exponential growth. In this case exp(-l1t)T(t)x converges to l1 P1x, which is a 1-dimensional projection of the information contained in the initial value x. Thus, the organization of the information in the initial value is de-synchronized over time.
Consider a population divided into 3 nonoverlapping age classes u1, u2, and u3:
where the 2nd age class produces 2 newborns and the 3rd age class produces 1 newborn. Each age class is subject to the logistic loss. Independent of crowding effects, 3/4 of the 1st age class survives to the 2nd age class and 2/3 of the 2nd age class survives to the 3rd age class.
The solutions for the 3 - age classes model for the initial values u1(0)=1, u2(0)=2, u3(0)=3. The solutions converge to equilibrium.
P1 is strictly positive on R3+ and its range is 1-dimensional. The parametric plot of exp(-l1t) exp(tA)x for various x = [x1, x2, x3]t is given below.
Parametric plot of the solutions of the Leslie age-structured model with logistic nonlinearity for various initial values [x1, x2, x3]t. Since the range of P1 is 1-dimensional, all solutions converge to a unique equilibrium.
The solution of the linear age-structured model is p(a,t) = (T(t)f)(a) for fin D(A). For arbitrary fin X, T(t)f is a generalized solution. Since P1 has 1-dimensional range, T(t), t > 0 has asynchronous exponential growth.
Theorems 1, 2, and 3 are applicable to the abstract formulation of this problem:
Telomeres are tandem repeats of DNA segments capping the ends of chromosomes. The “end-replication” hypothesis assumes that during mitosis incomplete replication of DNA occurs. Each time a cell divides the telomeres shorten in the subsequent generation and when a critical number is reached no further divisions occur. This critical number is known as the Hayflick limit and is estimated to be 40-80 in somatic vertebrate cells. The loss of telomeres is hypothesized as an explanation for the finite proliferative capacity of cell lines. Tumor stem cells are hypothesized to have the ability to regenerate their telomeres, and thus divide without limit.
Consider a proliferating cell population divided into n+1 classes of telomere lengths. Each time a cell divides the ends of its chromosomes, called telomeres, shorten. When a mother cell divides, one daughter cell has the same length as the mother and one daughter has a shorter length, except the 0 class, which does not divide. The telomere descent for n = 3.
The highest class u3(t) is constant in time, the next highest u2(t) grows linearly in time, the next highest u2(t) grows quadratically in time, and the lowest grows cubically in time. As time goes to infinity, the limit of the fraction of cells in the i th state,
ui(t) / ||u(t)||
depends only on the initial reservoir of cells in the highest state u3(t).
xi(0) = 0 for i = 0,…39, x40(0) = 100
xi(0) = 0 for i = 0,…39, x40(0) = 100, mi = .5
The logistic term accounts for increased mortality in each class dependent on the total population of all classes.
xi(0) = 0 for i = 0,…39, x40(0) = 106, mi = .3,t = 10-11
xi(0) = 0 for i = 0,…39, x40(0) = 102, mi = .3, i=0,..39, m40 =0,,t = 10-12
Theorems 4 and 5 are applicable to this model. Since l1 = 0, p(a,t) = (S(t)f )(a) converges to 0 for all f in X+.