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Outline

Age-dependent single-species population dynamicswith delayed argument

Antoni Leon Dawidowicz1 Anna Poskrobko2

1Institute of MathematicsJagiellonian University

2Faculty of Computer ScienceBialystok Technical University

Bia lystok 2008

A.L.Dawidowicz, A.Poskrobko Age-dependent single-species population dynamics

Outline

Outline

1 History of applications of mathematics in biology and medicine

2 von Foerster model

3 Gurtin and MacCamy model

4 Age-dependent model with delayed argumentStructureBiological justificationGlobal assumptionsGeneralization and precursorMain results

5 Conclusions


History of applications of mathematics in biology and medicinevon Foerster model

Gurtin and MacCamy modelAge-dependent model with delayed argument

Conclusions

History of applications of mathematics in biology andmedicine

1202 Fibonacci sequence (”Liber Abaci”)

1798 Malthus equation

1838 Verhulst equation

1926 Classical von Foerster model

1974 Gurtin and MacCamy model




Conclusions

von Foerster model

The first model applicable to age-dependent population dynamicswas proposed by von Foerster in 1926.

u(x , t)denote the density of a decomposition ofthe individuals in age x at time t;

z(t)the total population at time t denoted bythe formula:

z(t) =∫∞0 u(x , t)dx




Conclusions

von Foerster model

∂u

∂t+∂u

∂x= −λ(x)u(x , t)

u(x , 0) = v(x)

u(0, t) =

∫ ∞

0β(x)u(x , t)dx




Conclusions

von Foerster model

∂u

∂t+∂u

∂x= −λ(x)u(x , t)

u(x , 0) = v(x)

u(0, t) =

∫ ∞

0β(x)u(x , t)dx

The equation called in the literature as McKendrick equation or,more often, as von Foerster equation.




Conclusions

von Foerster model

∂u

∂t+∂u

∂x= −λ(x)u(x , t)

u(x , 0) = v(x)

u(0, t) =

∫ ∞

0β(x)u(x , t)dx

The model considers mortality.

λ(x)is called the death-modulus and describes the mortality perunit time individuals of age x




Conclusions

von Foerster model

∂u

∂t+∂u

∂x= −λ(x)u(x , t)

u(x , 0) = v(x)

u(0, t) =

∫ ∞

0β(x)u(x , t)dx

Initial condition




Conclusions

von Foerster model

∂u

∂t+∂u

∂x= −λ(x)u(x , t)

u(x , 0) = v(x)

u(0, t) =

∫ ∞

0β(x)u(x , t)dx

The birth process is described by the renewal equation.

β(x)birth-modulus, it is the average number of offspringsproduced (per unit time) by an individual of age x




Conclusions

Gurtin and MacCamy model

The model proposed by Gurtin and MacCamy (1974) was based onthe assumption that the progress of the population depends on itsnumber

∂u

∂t+∂u

∂x= −λ(x , z(t))u(x , t)

u(x , 0) = v(x)

u(0, t) =

∫ ∞

0β(x , z(t))u(x , t)dx

z(t) =

∫ ∞

0u(x , t)dx




Conclusions

StructureBiological justificationGlobal assumptionsGeneralization and precursorMain results

Age-dependent model with delayed argument

Our theory is based on the following system of the equations

∂u

∂t+∂u

∂x= −λ(x , zt)u(x , t)

z(t) =

∫ ∞

0u(x , t)dx

u(0, t) =

∫ ∞

0β(x , zt)u(x , t)dx

u(x , 0) = v(x),

wherezt : [−r , 0] → [0,∞), r > 0, zt = z(t + s)




Conclusions


Biological justification

It is common knowledge that other factors can have an influenceon the reproduction, for example:

a period of gestation;

a period of response of a system to stimulus.

These examples suggest the necessity to consider the descriptionswith delayed parameter. The delay is natural assumption in everybiological models, concerning for example problems ofepidemiology and immunology.




Conclusions


Global assumptions

(H1) ϕ ∈ L1(R+) is piecewise continuous;

(H2) λ, β ∈ C (R+ × C ([−r , 0])); the Frechet derivatives Dλ ofλ(x , ψ) and Dβ of β(x , ψ) with respect to ψ exist for allx > 0 and ψ > 0;

(H3) The functions λ(·, ψ), β(·, ψ) belong toC (C ([−r , 0]); L∞(R+));

(H4) The Frechet derivatives Dψ0λ and Dψ0β in the point ψ0 as afunction of ψ0 belong toC (C ([−r , 0]);L (C ([−r , 0]); L∞(R+))),where L(X ,Y ) denotes the Banach space of all boundedlinear operators from X to Y ;

(H5) ϕ > 0, λ > 0, β > 0.




Conclusions


Generalization and precursor

The elements which distinguish the model with delayed argumentfrom Gurtin nad MacCamy’s one:

the reproduction as the death depend on the population inany preceding period of time;

the dependence of λ and β on the variable z is functional one;

we consider the Frechet derivatives of the functions λ and βinstead of their partial derivatives.




Conclusions


Main results

Equivalent expression of the problem

Let u be a solution of the age-dependent population problem up totime T > 0. Then the population zt and the birth-rate B satisfyon [0,T ] the operator equations

zt(s) =

∫ t+s

0B(x)e−

R t+sx λ(τ−x ,zτ )dτdx (1)

+

∫ ∞

0ϕ(x)e−

R t+s0 λ(x ,zτ )dτdx

and B(t) =

∫ t

0β(t − x , zt)B(x)e−

R tx λ(τ−x ,zτ )dτdx (2)

+

∫ ∞

0β(x + t, zt)ϕ(x)e−

R t0 λ(x+τ,zτ )dτdx .




Conclusions


Main results

Equivalent expression of the problem, continuation

Conversely, if zt and B are non-negative continuous functionssatisfying (1) and (2) on [0,T ], and if u is defined on R+ × [0,T ]by the formula

u(x , t) =

{ϕ(x − t)e−

R t0 λ(x−t+τ,zτ )dτ for x > t

B(t − x)e−R x0 λ(α,zt−x+α)dα for t > x

,

then u is a solution of the age-dependent population problem up totime T .




Conclusions


Main results

B(t) =

∫ t

0β(t − x , zt)B(x)e−

R tx λ(τ−x ,zτ )dτdx

+

∫ ∞

0β(x + t, zt)ϕ(x)e−

R t0 λ(x+τ,zτ )dτdx .

ZT (z)(s) =

∫ t+s

0BT (z)(x)e−

R t+sx λ(τ−x ,zτ )dτdx

+

∫ ∞

0ϕ(a)e−

R t+s0 λ(a+τ,zτ )dτda.




Conclusions


Main results

B(t) = BT (z)(t)

ZT (z)(s) =

∫ t+s

0BT (z)(x)e−


+

∫ ∞

0ϕ(a)e−





Conclusions


Main results

Lemma

There exists T > 0 such that the operatorZT : C+[−r ,T ] → C+[−r ,T ] defined by

ZT (z)(s) =

∫ t+s

0BT (z)(x)e−


+

∫ ∞

0ϕ(a)e−


has a unique fixed point.




Conclusions


Main results

Local existence of the solution

There exists T > 0 such that the population problem has a uniquesolution up to time T .

Global existence of the solution

If the average number of offsprings (per unit time) β(x , zt) isuniformly bounded for all x and zt , i.e. β = sup x>0

zt>0β(x , zt) <∞,

then the age-dependent population problem has a unique solutionfor all time.

Stability of the equilibrium age distribution




Conclusions

Conclusions

Von Foerster modelthe dependence of the populationdynamics on age;

Gurtin-MacCamy modelthe reproduction as the death dependon the number of the population;

The above model

functional dependence of the birthand death moduli on the populationin any preceding period of time.
