II. GENERAL THEORY OF AGE-STRUCTURED POPULATION DYNAMICS

1. 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

2. 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

3. 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)

5. 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.

6. 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+ .

7. For the classical scalar logistic equation X = R, Ax = l1 x,Fx = x/C, T(t)x = exp(l1 t) x, and

8. 3. Asymptotic behavior of the abstract logistic equation

9. 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.

10. 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.

11. 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.

12. The abstract version of this system of equations is

13. The hypotheses of Theorems 1 and 2 are satisfied with

14. 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.

15. 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.

16. 5. A linear continuous age-structured cell population model

18. 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.

19. 6. Nonlinear continuous age-structured population models Theorems 1, 2, and 3 are applicable to the abstract formulation of this problem:

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

22. 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.

23. 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.

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

25. 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).

26. 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

27. Linear ODE model of telomere shortening with mortality in each class

28. 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

29. 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.

30. This system can be written in vector form as

31. 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

32. The phase portrait of the solutions u10(t), u20(t), and u30(t)

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

34. 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

35. The phase portrait of the solutions u10(t), u20(t), and u30(t)

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

37. Semigroup formulation of the linear age-structured telomere model:

39. 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+.

41. References