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II. GENERAL THEORY OF AGE-STRUCTURED POPULATION DYNAMICS

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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

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)

The classical logistic equation can generalized to the form

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.

2. The abstract logistic equation in a Banach Lattice

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+ .

For the classical scalar logistic equation

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.

4. The Leslie age-structured model with logistic nonlinearity

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 semigroup formulation of the age-structured model:

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.

6. Nonlinear continuous age-structured population models

Theorems 1, 2, and 3 are applicable to the abstract formulation of this problem:

7. The abstract logistic equation when the inherent linear growth rate is polynomial in time

8. An application to telomere shortening in cell populations

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.

A linear ODE model of telomere loss in proliferating cell population lines

This semigroup has polynomial growth in the following sense:

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).

An example of the linear ODE telomere model with n=40 and no mortality in any telomere class

xi(0) = 0 for i = 0,…39, x40(0) = 100

An example of the linear ODE telomere model with n = 40 and mortality in each class

xi(0) = 0 for i = 0,…39, x40(0) = 100, mi = .5

A nonlinear ODE model of telomere shortening with logistic mortalityin every telomere class

The logistic term accounts for increased mortality in each class dependent on the total population of all classes.

An example for the nonlinear ODE telomere model with n=40 and logistic nonlinearity in every telomere class

xi(0) = 0 for i = 0,…39, x40(0) = 106, mi = .3,t = 10-11

A nonlinear ODE model of telomere shortening with no mortality in the highest telomere class

An example for the nonlinear logistic ODE model with n=40 and with no mortality in the highest telomere class

xi(0) = 0 for i = 0,…39, x40(0) = 102, mi = .3, i=0,..39, m40 =0,,t = 10-12

9. Telomere shortening in a linear continuous age-structured cell population model

10. Telomere shortening in a nonlinear continuous age-structured cell population model

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+.

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