Population Models with Age and Space Structure

Thesis

Population Models with Age and Space

Structure

Anton Karlsson

LiTH-MAT-EX2017/03SE

Population Models with Age and Space Structure

Department of Mathematics, Linköping University Anton Karlsson

LiTH-MAT-EX2017/03SE

Thesis: 16 hp Level: G2

Supervisor: Bengt-Ove Turesson,

Department of Mathematics, Linköping University Examiner: Bengt-Ove Turesson,

Department of Mathematics, Linköping University Linköping: March 2017

Abstract

In this thesis, basic concepts of populational models are studied from a theoret-ical point of view, especially the long term behaviours. All models are at least time dependent with additional age structure, spatial structure. The last model which is an extension of the von Foerster equation, is dependent on all o f these structures and have a long-term solution for large values of time. Modeling pop-ulation is a frequent subject in modern biology. It is hard to create a model that appears as realistic as possible. First one might consider that a population size is governed by the current size of the population, along with rates of how each individual contributes (give birth), so that the population increases. and how frequent an individual dies, causing the population to decrease in size. However these sort of models can only describe the size of population in a shorter span of time.

Keywords:

Mathematical Biology, Partial Dierential Equations, Long-term Behaviour. URL for electronic version:

urn:nbn:se:liu:diva-134926

Acknowledgements

First i like to thank my supervisor Dr. Bengt-Ove Turesson for all the help and guidance throughout this thesis.

I would also like thank my opponent and dear friend Karl Rönnby who has given constructive critic and has been supportive thought this thesis.

A special thanks to Martin Brengdahl and Inger-Marie Wolfarth for inter-esting discussions on the more practical side of the subject.

Lastly, I like to thank Olle Abrahamsson for giving me some helpful advice through out this thesis.

Letters

t ∈ R, usually ≥ 0 time variable

x ∈ R Spatial coordinate

a ∈ R, usually ≥ 0 age variable

n = n(t, a)or n(t) ∈ R population size

u = u(x, t, a) ∈ R Population distribution

Symbols

Ω ⊆ Rn _{spatial area of where population grows.}

R The set of all real numbers

Rn n-dimensional Euclidean space

d· dx = (·)

0 _{dierentiation over a variable x} ∂·

∂x= (·)x partial dierentiation over a variable x.

∆ =Pn

i=1 ∂2 ∂x2

i The Laplacian operator on the spatial variable x = (x1

, . . . , xn) ∈ Rn

Contents

1 Dierential equations 1

1.1 Equilibrium points . . . 1

1.2 Nondimensionalisation . . . 2

1.3 Characteristic curves . . . 3

1.4 Travelling wave solutions. . . 4

2 Some basic populational models 7 2.1 The balance Law . . . 7

2.2 A rst model . . . 8

2.3 The logistic equation . . . 9

2.4 Delay models . . . 10

2.5 The Lotka-Volterra model . . . 12

3 Models with age structure 17 3.1 The von Foerster equation . . . 17

3.2 Long-term solution of the von Foerster equation. . . 20

4 Models with spatial structure 23 4.1 The Fisher equation . . . 23

5 Age dependent models with spatial structure 27 5.1 An extension of the von Foerster equation . . . 27

5.2 Separation of variables of equation (5.5) . . . 29

5.3 Solution of equation (5.14). . . 31

5.4 Some nal remarks . . . 33

A Phase plane diagram and stability 35 B Method of separation for Langlais equation 39 C Functional Analysis 41 C.1 Normed spaces . . . 41

C.2 Inner product spaces . . . 42

C.3 Sobolev spaces . . . 43

List of Figures

1.1 Equilibrium points for (1.2). . . 3

2.1 Some solutions for the equation (2.4). . . 9

2.2 Two solutions for (2.5) showing visually that n = K is stable. . . 10

2.3 A graph showing how the prey population u(τ) is followed by the prey population. . . 13

2.4 The phase plane for the Lotka-Volterra model. Here we see how the solutions behave in circular fashion around (u, v) = (1, 1), and disperse from (u, v) = (0, 0) in the rst quadrant. . . 14

A.1 Three dierent equilibrium points. . . 38

Chapter 1

Dierential equations

This thesis is based around dierential equations and partial dierential equa-tions in particular. In this chapter, we go through how these equaequa-tions are analysed and also cover something about how biological models are handled and interpreted. In the beginning of this chapter we study for the most part or-dinary dierential equations (ODE), which are based on one variable, and later introduce partial dierential equations (PDE), which are dependent on more than one variable.

1.1 Equilibrium points

It is common to solve an equation analytically, regardless whether the equation is an ODE or a PDE. One way is to nd a solution numerically (which is not covered in this thesis). However, we could also analyse the solutions of an equation and see how they behave in the long-term. For example we could see what would happens when a time variable, t → ∞. We might nd that the solution will be bounded or that it will be unbounded. We could also see how a solution would be aected by the initial conditions this way. Take the following ODE known as Newtons cool-down equation [4, p. 40]:

dx

dt = −k(x − A), t > 0 (1.1)

for k > 0 and an arbitrary constant A. With the initial condition x(0) = x0, we

can nd a solution by separation of variables: x = A + De−kt,

where D = eC _{and C is an arbitrary constant. With x(0) = x}

0we have that

x(t) = A + (x0− A)e−kt, t ≥ 0.

We can see that x(t) tends to A when t → ∞. Another way of saying this is that x(t) tends to the solution with the boundary condition x0= 0, i.e., x(t) = A

for all t. We call the solution x(t) = A a critical point or an equilibrium point. To be more precise, we will use the following denition:

Denition 1.1.1. We call a solution x = c to a dierential equation dx

dt = f (x) an equilibrium point if f(c) = 0.

In Equation (1.1), we can see that the x(t) = A is an (unique) equilibrium point for (1.1). We also saw that solutions for all initial conditions will tend to Awhen t tends to innity. This is an example of a so called stable equilibrium point. In general, a stable equilibrium point satises the following denition. Denition 1.1.2. An equilibrium point x = c for a dierential equation

dx

dt = f (x),

is called stable if there exists some solution x(t) such that x(t) → c

as t → ∞. Otherwise we say that x(t) is an unstable equilibrium point. Example 1.1.3. Take the following logistic dierential equation.

dx

dt = kx(M − x), (1.2)

which is a Bernoulli equation [8, p. 79-80], so it has the solution x(t) = M e

kM t

ekM t_{+ M C}, (1.3)

for some constant C ∈ R. Equation (1.2) has the equilibrium points x = 0 and x = M. Figure1.1describes a solution x(t), with initial condition x(0) = x0.

When x0> M, the solution x(t) will tend to x(t) = M, as t tend to ∞, and x(t)

will tend to x(t) = 0 as t tend to −∞. This indicated by the blue line in Figure

1.1. The interested reader can study Figure 1.1 for other initial conditions as well.

1.2 Nondimensionalisation

When it comes to modeling physical problems, such as population models, there are models with physical properties such as time, coordinates, etc. So in order to study such models we strip down these properties to a dimensionless state. We shall see that by nondimensionalise an equation, we are studying the relations between the parameters rather then the parameters themselves. As Murray (1984) wrote: Before analysing the model it is essential, or rather obligatory, to express it in nondimensional terms. [7, p. 7]. Take the following popula-tional model with a delay (more on delay models in Section2.4):

dn dt = rn(T )  1 −n(t − T ) K  , t > 0, (1.4)

1.3. Characteristic curves 3 M t x(t) _x 0> M x0< 0 0 < x0< M

Figure 1.1: Equilibrium points for (1.2).

where r, T and K are three constants. t and T are of dimension time1_{, n is}

of dimension size of population population1_{, r is time}−1_{, and K represents}

the capacity of an environment for a population (such as food), so just as n its dimension is population1_{. We can now choose new terms such that they cancel}

out the dimensions of the original terms, namely t∗= rt T∗= rT n∗= n/K.

Replace all variables and parameters of (1.4) with the new dimensionless ones. Dierentiation on both sides of (1.4) and the chain rule yields

rKdn ∗ dt∗ = rKn ∗_{(1 − n}∗_(t∗_{− T}∗_{)) ,} that is, dn∗ dt∗ = n ∗_{(1 − n}∗_(t∗_{− T}∗_{)) .}

As a result, we have less terms in our equation and the new terms can be seen as the relation of the old terms, say if n larger than 1, then n is larger than K.

1.3 Characteristic curves

When it comes to solving linear partial dierential equations, a method often used is characteristic curves. What it does is nding curves along which the solution exists. An important application is that by using characteristic curves we can take a PDE and reduce it to an ODE.

Example 1.3.1. Consider the PDE

aux+ buy= 0 (1.5)

for a function u(x, y) : R × R → R, where a and b are two constants. Let v be a vector such that

a b 

• v = 0,

where • denotes the dot product. That is, v is orthogonal to (a, b)T_{, which}

yields that v = (b, −a)T_{. This vector is of the normal of any line of the form.}

bx − ay = c,

where c ∈ R. Let f : R → R be a C1_{function. Then a solution to (1.5) is given}

by u(x, y) = f(c) = f(bx − ax), since

aux+ buy= 0,

becomes

a(bf0) + b(−af0) = abf0− abf0 = 0.

The lines bx − ay = c are often called the characteristic curves or the charac-teristics for the equation (1.5).

The example above only illustrated a particular case of nding characteristic curves. It gets more tricky to nd these as we shall see in Chapter3. The general idea, however, is to nd curves where the equations behaves in way which are easy to work with.

1.4 Travelling wave solutions

A phenomena that occurs often in nature is waves. Here we will study travel-ling waves which behaves in a wave-like pattern over time. We shall use the following denition for travelling waves.

Denition 1.4.1. Let x ∈ Rn _{and t ∈ R. A travelling wave is a function of}

the form

u(x, t) = u(x − ct) (1.6)

where c > 0 is a constant.

Remark 1.4.2. Note that if a PDE has a travelling wave solution u(x − ct), then it is also a characteristic curve x − ct.

Example 1.4.3. The following example is found in [9, p. 33]. A classic result in which travelling solutions appear is in the wave equation:

∂2_u

∂t2 = c 2∂2u

∂x2, (x, t) ∈ R 2_,

which has the solution

u(x, t) = f (x + ct) + g(x − ct),

where f, g ∈ C2_{. This solution is interesting since it has two travelling}

waves; f(x + ct) travells to the left, as t gets larger, and g(x − ct) travells to the right as t gets larger.

1.4. Travelling wave solutions 5

A common area that travelling wave solutions exist in is diusion. Adolf Fick formulated two laws for diusion of a substance. The second law states for a density u(x, t), whether it be for a gas, a population, etc., that [9, p. 15]

∂u ∂t = D

∂2_u

∂x2, (1.7)

where D is the diusion constant and its dimension being length2_time−1 _[2].

In Chapter4, we will add a term, f(u), to (1.7), so it becomes ∂u

∂t = f (u) + D ∂2_u

∂x2. (1.8)

We will study a special case of (1.8) in Chapter4, which will exhibit a travelling wave solution.

Chapter 2

Some basic populational

models

In this chapter, we will introduce some common dierential equations that are modeled towards populations. First we will introduce the balance law, which formulates the general behaviour of populations

2.1 The balance Law

Here we will cover a law that will be used in some models later in the thesis. According to Ellner and Guckenheimer [5, p. 32], we have that for a popula-tion N(t), of discrete time t,

N (t + 1) = N (t) +Birth + Immigration − Death − Emigration, which we can rewrite as

N (t + 1) − N (t) =Birth + Immigration − Death − Emigration. (2.1) We see that the balance law describes how the population will change at next time step. Here, Birth, Death, Immigration and Emigration are functions. In simple cases, we assume that these are constants (as we will see in Section2.2). We will refer to the sum on the right-hand side of (2.1) as BIDE.

However, this thesis does not consider populational models with discrete variables, therefore we need to extend (2.1) to the continuous case. That is, with the change of a small increment of time, t. To clarify, replace the 1 in (2.1) with an h ∈ R, and replace also the function N(t) with a function n(t), where n(t) is an extension of N(t) into continuous variables. Then for the change for a time t to t + h is given by the balance law as

n(t + h) = (b + i)hn(t) − (µ + e)hn(t), and so,

n(t + h) − n(t)

h = (b + i)n(t) − (µ + e)n(t), (2.2)

where b, i, µ, e, denotes the terms birth, immigration, death and emigration, respectively. Then by letting h → 0, (2.2) becomes

dn

dt = (b + i)n(t) − (µ + e)n(t), (2.3)

We will see this law being applied in Chapter2.2where we will let Immigration = Emigration = 0, and Birth and Death be dependent on the size of the popula-tions.

2.2 A rst model

A populational model does not have to be that complex. We can create a model for a population n only determined by two factors: birthrate B and death rate D, both dependent of time t ∈ R. That is,

dn

dt = (B(t) − D(t)) n, t > 0. (2.4)

If we let B(t) = b and D(t) = µ, where b and µ ∈ R are two constants, then (2.4) becomes

dN

dt = (b − µ)N, t > 0.

The solution to this model is simply

N (t) = N0e(b−µ)t, t > 0,

where N0 ∈ R is the initial population at t = 0. This might not be the best

way to model a population. However it can be of use in describing populations in very short time intervals. If we let either b > µ or µ > b, we can see how the solution will grow or decay exponentially. As Figure2.1shows, the population will grow exponentially fast when the birthrate is larger than the death rate, and the population dies out fast when the death rate is larger than the birthrate. The population will only be steady when b = µ; then the population will be constant equal to N0 for all t.

2.3. The logistic equation 9 0 N0 t N (t) b > µ b < µ b = µ

Figure 2.1: Some solutions for the equation (2.4).

2.3 The logistic equation

There are ways to model a population so that it neither grows nor decays ex-ponentially as the model presented in Section 2.2 did. One way is to add a resource factor K > 0. The idea is that when a population gets to large for the resources, the population will decline, and vice versa for when the population is small. A way to model this was suggested by Verhulst [7, p. 3]:

dn

dt = rn(t) (1 − n(t)/K) , , t > 0 (2.5) where r and K are two positive constants. Without calculating we can see that the equations have two equilibrium points, n = K and n = 0. We can see that n = K is stable if n(0) < K. Then

dn dt > 0 for small increments of t. We have that dn

dt > 0for all n < K so n will increase

to n = K; say that this happens at t = T . After that we have dn

dt = 0

for all t > T . We can in the same way see how the solutions will behave if we let n(0) > K. The solution is found by realising that (2.5) is a Bernoulli equation, so it has the solution

n(t) = Kre

rt

0 K

t n(t)

Figure 2.2: Two solutions for (2.5) showing visually that n = K is stable. for some constant C. With an initial condition n(0) = n0, the following

expres-sion for C is given

C =Kr − n0 Krn0

, Krn06= 0.

We can plot the following solutions in Figure2.2.

2.4 Delay models

An occurring type of model in biology are delay models as they exhibit oscillating solutions which are similar to that of biological populations.

Denition 2.4.1. A delay model is of the form dn

dt = f (n(t), n(t − T )), t > 0, for some constant T and some function f : R × R → R.

We can see how this sort of model can have oscillating solutions. Take (2.5) and adjust it into a delay model

dn dt = rn(t)  1 −n(t − T ) K  (2.6) where, as in (2.5), r and K are two positive constants. Now, let t1 and t2 be

two constant such that t1 < t2, and let n(t1) = n(t2) = K and for t < t1,

let n(t − T ) < K. This implies that 

1 −n(t − T ) K

 > 0,

2.4. Delay models 11

and so

dn(t) dt > 0,

that is n(t) is increasing for t < t1. If we let t increase so that t = t1+ K, then

n(t − T ) = n(t1) = K,

so

dn(t) dt = 0. Then if t1+ T < t < t2, we get that

n(t − T ) > K, which yields

dn(t) dt < 0, and nally we have that

n(t2+ T − T ) = n(t2) = K,

which gives us that

dn(t2)

dt = 0

Thus we see that n(t) behaves in an oscillatory manner if we continue in this fashion.

One might wonder if a model that is not a delay model can behave in this periodic way. It is easy to show that this is not possible.

Theorem 2.4.1. The dierential equation dn

dt = f (n), t ∈ R,

where f ∈ C1_{, has a periodic solution, n(t), if and only if, n(t) is constant.}

Proof. We want to show that a solution n(t) is not periodic, that is for any T , n(t + T ) 6= n(t) for all t ∈ I. Take the integral

Z a+T a  dn dt 2 dt = Z a+T a dn dt dn dt dt = Z n(a+T ) n(a) dn dt dn = Z N (a+T ) n(a) f (n) dn = F (n(a + T )) − F (n(a)) = F (n(a)) − F (n(a)) = 0,

where F (n) is the primitive function of f(n). Thus we conclude that n0 _{= 0for}

2.5 The Lotka-Volterra model

Another big topic in the eld of populational models is the study of relations between two or more populations. In this case we consider two populations: a prey population and a predator population. The idea is that when the prey population is large, the predators will have resources to grow until the prey population has declined to a point were there are not enough prey left the the entire predator population. Then the predator population will decrease, which in turn yields that the prey population will grow, and so on.

The following model was proposed by Volterra [7, pp.79-80], with the prey n = n(t)and predator p = p(t), (_dn dt = n(a − bp) dp dt = p(cn − d) (2.7) where a, b, c, and d are positive constants. Murray points out these important assumptions that the model is based upon [7, p.80].

1. The an term represents how the prey population will grow in absence of the prey. Indeed it will grow in a rapid way, similar to (2.4), with a low death rate.

2. The −bnp term is the rate by which the prey is reduced by the predator. 3. The −dp term describes how the predator will exponentially decay, similar

to (2.4), with a high death rate.

4. The cnp term is the preys contribution to the predator.

Before we do any analysis of this equation, we will reduce (2.7) to a dimensionless state by dimensionalaising (see Section1.2). Let

u(τ ) = cn(t) d , v(τ ) = bp(t) a , τ = at, α = d a. (2.8)

Then we have that from the left hand side of the rst part of (2.7) dn(t) dt = d dt du(τ ) c = d c du(τ ) dτ dτ dt = a d c du(τ ) dτ . From the right hand side of (2.7) we have that

n(t) (a − bP (t)) = d cu(τ )  a − ba bv(τ )  = ad cu(τ ) (1 − v(τ )) . Thus, ad c du dτ = a d cu(τ ) (1 − v(τ )) , if and only if,

du

dτ = u(τ ) (1 − v(τ )) . (2.9)

For the second equation of (2.7) we have with similiar calculations dv

2.5. The Lotka-Volterra model 13

With (2.9) and (2.10), we have the new ODE system (_du

dτ = u(τ ) (1 − v(τ )) dv

dτ = αv(τ ) (u(τ ) − 1) .

(2.11) We have two singular points: (u, v) = (0, 0), and (u, v) = (1, 1). The stability of these equilibrium points are decided by their eigenvalues. In order to nd the eigenvalues we shall use a technique called linearisation (see AppendixA). Dene a function f(u, v) : C1_{× C}1

→ R2 _{such that}

f (u, v) = u (1 − v) αv(u − 1)



The we see of that

Jf(u, v) = 1 − v −u α α(u − 1)  . Then for Jf(1, 1) =  0 −1 α 0  (2.12) which has the eigenvalues λ = ±iα with the corresponding eigenvectors x = (1, ∓i√α). This yields a solution

u(τ ) v(τ )  =  1 −i√α  ei √ ατ ₊  1 i√α  e−i √ ατ_{, (2.13)}

for small τ and with boundary conditions that are close to the singular point (u, v) = (1, 1). We can see that this solution is circular in a counter clockwise direction of period τ = [0,2π

α].This behaviour indicates that the prey population

is followed by the predator populations, as indicated by Figure2.3.

0 0

τ

u(τ ) v(τ )

Figure 2.3: A graph showing how the prey population u(τ) is followed by the prey population.

0 1 0

1

Figure 2.4: The phase plane for the Lotka-Volterra model. Here we see how the solutions behave in circular fashion around (u, v) = (1, 1), and disperse from (u, v) = (0, 0)in the rst quadrant.

The equilibrium at (u, v) = (1, 1) is arguably the most interesting equilib-rium point since both populations will circle around it. If we had assumed that the boundary condition (u(0), v(0)) = (1, 1) then the solutions would always be (u(τ ), v(τ )) = (1, 1) for all τ. However, the populations will draw away from the other equilibrium point, (0, 0).

At the equilibrium (u, v) = (0, 0) we have that the Jacobian, Jf, becomes

Jf =

1 0

0 −α 

(2.14) which yields the eigenvalues λ = 1, −α, corresponding to an unstable saddle point, since the eigenvalues are real and does not have matching sign. We can see how dierent solutions will behave around the equilibriums in Figure2.4.

What about the solution for (2.11)? We can nd a solution by studying the phase trajectories

dv du =

u(τ ) (1 − v(τ )) αv(τ ) (u(τ ) − 1),

2.5. The Lotka-Volterra model 15

which by integration with respect to u, we nd that αu + v − ln uαv = H,

where H > Hmin= 1 + α[7, p. 80]. The reader can also observe these solution

Chapter 3

Models with age structure

This far, in the models that we have studied, we have only made use of one variable, t, which has represented time. We are now adding an extra dimension, namely the age of an individual, which will be denoted by a. We will have solutions of the form n(t, a), so to measure the size of the entire population at a desired time t = T is dened by

n(T ) = Z ∞

0

n(T, a) da, that is, the sum of all individuals at time T .

3.1 The von Foerster equation

We shall now introduce a new equation. Let n(t, a) be the population density that is dependent of both time t and age a. This model will not account for any migration, nor emigration, so according to the balance law, from Section 2.1, only the populations birth and death process will aect its size.

Let µ(a)n(t, a)dt be the rate for which the population changes for a small increment, dt, of time t. An individual of the population enters the population only at time a = 0, that is at birth. This means mathematically that the birth process will not be in the equation dened below, however it will appear in the boundary condition for n(t, 0), which will be dened below. The conservation law now says that

dn(t, a) = ∂n ∂tdt +

∂n

∂ada = −µ(a)n(t, a)dt. (3.1) We can use the fact that age changes at the same rate as time, that is da = dt, if and only if da/dt = 1. Then by dividing (3.1) by dt, we have

∂n ∂t +

∂n

∂a = −µ(a)n(t, a), t, a > 0, (3.2) which is known as the von Foerster equation [7, p. 37 ]. Furthermore we will add two boundary conditions. Starting by adding one condition for the initial time t = 0, we have that

n(0, a) = f (a), a > 0, (3.3)

where f(a) is a given age distribution. As mentioned above, birth can only occur at a = 0, so in this condition we will use the birth function:

n(t, 0) = Z ∞

0

b(a)n(t, a) da, t > 0, (3.4) that is, the total contribution made by individuals from the age span 0 ≤ a < ∞. Of course, the number of births, b(a), will tend to zero as a → ∞.

Later, we can discuss what will happen to the population by choosing certain birth and death rate functions. But rst, we shall look at the solution.

We are going to solve (3.2) by using the characteristics or characteristic curves of the equation, which we learned about in Section 1.3. The solution satises the integral equation in the following theorem.

Theorem 3.1.1. A solution, n(t, a) of (3.2) is given by n(t, a) = f (a − t) exp  − Z a a−t µ(σ) dσ  , a > t, and n(t, a) = n(t − a, 0) exp  − Z a 0 µ(σ) dσ  , a < t, where n(t − a, 0) is given by the linear integral equation

n(t, 0) = Z t 0 b(a)n(t − a, 0) exp  − Z a 0 µ(σ) dσ  da + Z ∞ t b(a)n(t − a, a) exp  − Z a a−t µ(σ) dσ  da. (3.5) Proof. Let s be a real number on which t and a are parameterised on, so that

n(t, a) = n (t(s), a(s)) = n(s). From (3.2) and by the chain rule, we get

−µ(a(s))n(s) = dn(s) ds = ∂n ∂tt 0_{(s) +}∂n ∂aa 0_(s).

Comparing this to (3.2), we get that

t0(s) = a0(s) = 1, (3.6)

and an ODE,

n0(s) = −µ(a(s))n(s),

which is a rst-order linear ODE, so its solution is found by using integrating factors, and thus

n(s) = n0exp  − Z s 0 µ(σ) dσ  , and by making the substitution τ = σ + a0,

n(s) = n0exp  − Z a+a0 a0 µ(τ ) dτ  . (3.7)

3.1. The von Foerster equation 19

Let t0 and a0be positive real numbers, so that the solutions of (3.6) is given by

(

t = s + t0,

a = s + a0,

which are the characteristic curves for (3.2). Now, consider the t, a-plane and study the line t = a. For the solutions above the line, that is, when a > t, we set t0= 0, and so

a = t + a0, t = s.

Plugging this into (3.7), we get that n(t, a) = n(0, a0) exp  − Z s+a0 a0 µ(σ) dσ  , a > t, and with the initial condition (3.3) and using that a0= a − t,

n(t, a) = f (a − t) exp  − Z a a−t µ(σ) dσ  , a > t.

By similar reasoning for solution below the line t = a, that is when, a < t, we let a0= 0, and so a = t − t0, s = a, which in (3.7) yields, n(t, a) = n(t0, 0) exp  − Z a 0 µ(σ) dσ  , and with t0= t − abecomes,

n(t, a) = n(t − a, 0) exp  − Z a 0 µ(σ) dσ  ,

where n(t−a, 0) is given by the condition (3.4), and using the solution for n(t, a) when a < t and when a > t:

n(t, 0) = Z t 0 b(a)n(t − a, a) exp  − Z a 0 µ(σ) dσ  da + Z ∞ t b(a)f (a − t) exp  − Z a a−t µ(σ) dσ  da.

The integral equation for n(t − a, 0), has a solution that can be obtained with iteration, according to Murray in [7, p. 38]. This is however out of the scope for this thesis. We are rather interested in how a population will behave for large t. We will study this in the next section, and we will use those results in the later part of thesis in Chapter5.

According to Murray, t0 and a0, which was mentioned in the proof above,

can be interpreted respectively as the time of birth of an individual, and the initial age of an individual at time t = 0 [7, p. 37].

Finally, the solution n(t, a) for a > t represents the population before time t = 0 and the other solution, when a < t represents the population after time t = 0.

3.2 Long-term solution of the von Foerster

equa-tion

Since we cannot write the solution n(t, a) in an explicit form, we can instead study the long-term behaviour of the solution, that is when t → ∞. This way, a simpler result can be attained. Assuming that t is large and so that t > a, we are only interested in the solution for (3.2) for t > a, which is given by Theorem3.1.1as n(t, a) = n(t − a, 0) exp  − Z a 0 µ(s) ds  da,

where the second integral in the integral equation for n(t − a) tends to zero. However, this does not satisfy the initial condition for a at (3.4) (see [7, p. 38]). We can look for solutions of the form

n(t, a) = eγtr(a), t, a > 0 (3.8) where γ is a constant.

Theorem 3.2.1. A solution, n(t, a), to (3.2) of the form (3.8) is given by n(t, a) = eγtr(a), t, a > 0, where r(a) = r(0) exp  −γa − Z a 0 µ(s) ds  , a > 0, and γ is such that

1 = Z ∞ 0 b(a) exp  −γa − Z a 0 µ(s) ds  da. (3.9)

Proof. We can use (3.8) in the original equation (3.2). An equation for r(a) can then be derived: ∂n ∂t + ∂n ∂a = ∂ ∂te γt_{r(a) +} ∂ ∂ae γt_r(a)

= γeγtr(a) + eγtr0(a), and with ∂n ∂t + ∂n ∂a = −µn, we get that

γeγtr(a) + eγtr0(a) = −µeγtr(a). Dierentiate r(a):

r0(a) = −µ(a)eγtr(a) − γeγtr(a)
e−γtr(a) = −(µ(a) + γ)r(a).

Then the solution for r(a) is given by r(a) = r(0) exp  −γa − Z a 0 µ(s) ds  . (3.10)

3.2. Long-term solution of the von Foerster equation 21

So, with the initial condition (3.4), n(t, 0) = eγtr(0) = Z ∞ 0 b(a)eγtr(0) exp  −γa − Z a 0 µ(s) ds  . Dividing both sides with eγt_r(0),

1 = Z ∞ 0 b(a) exp  −γa − Z ∞ 0 µ(s) ds  .

Although this solution yields an another unknown r(0), we are mainly in-trested in the long term behaviour ohow the population, so this will not be im-portant. What is more intresting is the thereshold of the population S, which will determine wether the population will grow or decay.

Chapter 4

Models with spatial structure

In this chapter, the age structure is replaced by a spatial structure. Our focus will be how a population will disperse in the spatial domain.

4.1 The Fisher equation

In Section1.4, we saw that diusion can be described by Fick's second law: ∂u

∂t = D ∂2u

∂x2, (4.1)

where D is a positive diusion constant. We can expand this into an equation of the form

∂u

∂t = f (u) + D ∂2_u

∂x2. (4.2)

One of the simplest of such equations is the Fisher-Kolmogorov equation, which we get when f(u) = ku(1 − u):

∂u

∂t = ku(1 − u) + D ∂2_u

∂x2, (4.3)

where k is a positive constant [7, p. 440]. Before analysing, we will make a nondimensionalisation.

Theorem 4.1.1. Equation (4.3) can be written in the nondimensional equation ∂u

∂t = u(1 − u) + ∂2_u

∂x2. (4.4)

Proof. Introduce the nondimensional terms x∗= _Dk
1/2

t∗= kt, so that the right-hand side of (4.3) becomes

∂u ∂t = ∂u ∂t∗ ∂t∗ ∂t = k ∂u ∂t∗ Karlsson, 2017. 23

and the on the right-hand side of (4.3) becomes D∂ 2_u ∂x2 = D ∂ ∂x∗ ∂x∗ ∂x  ∂u ∂x∗ ∂x∗ ∂x  = D k D  ∂2_u ∂x2 = k ∂2_u ∂(x∗₎2.

Thus, the nondimensionalisation transforms (4.3) into ∂u

∂t = u(1 − u) + ∂2_u

∂x2.

We shall see that Equation (4.4) will exhibit a travelling wave solution, that is u is of the form u = f(z) where z = x − ct, where c is a constant representing the wave speed. When we let u(t, x) = f(x − ct), for some f ∈ C2_{, we see}

that (4.4) can be written as

f00+ cf0+ f (1 − f ) = 0, (4.5) where the prime denotes dierentiation with respect to z. We can see that a travelling wave solution exists, according to the following theorem.

Theorem 4.1.2. For a solution u(x, t) = f(x−ct) of (4.4), the minimum value for c is 2. The solution f(z) have the equilibrium points f = 0 and f = 1.

The following proof uses what is called phase plane analysis. The reader can nd more details in AppendixA.

Proof. We are going to analyse a phase plane, where the function f will be compared to g = f0_{. We have that}

f0_{= g}

g0 = −cg − f (1 − f ). Thus, the phase plane trajectories are solutions of

g0 f0  =−cg − f (1 − f ) g  . (4.6)

There are two equilibrium points to (4.6), namely (f, g) = (0, 0) and (1, 0). The stability is given by the eigenvalues of Equation (4.6) around these points (see Appendix A). For (f, g) = (0, 0), we have

λ = −1 2 h

c ± c2− 4
1/2i

which means that (0, 0) is a stable node for c2_{> 4and a stable spiral for c}2_{< 4,}

thus if a solution f is close to 0, it will oscillate for a wave speed c2 _{< 4. The}

equilibrium at (1, 0) is a saddle point which is neither stable nor unstable. This is because it has both a positive and a negative eigenvalue:

λ = −1 2 h

c ± c2+ 4
1/2i .

From this we can deduce that the wave speed c is at least cmin= 2. In the case

4.1. The Fisher equation 25

Now we can study what sorts of initial conditions will exhibit a travelling wave solution. According to Murray [7, p. 442], it has been worked out by Kolmogorov that if the boundary for t = 0,

u(x, 0) = u0(x) ≥ 0, u0(x) =

(

1 if x ≤ x1

0 if x ≥ x2

where x1 < x2 and u0(x) is continuous on the open interval (x1, x2), then the

solution u(x, t) is a travelling wave solution of the form f(x − 2t). That is, it exhibits the lowest wave speed possible. It is also important to see what happens to the solution as x → ∞. We shall do this by linearise (4.4), so that we have

∂u ∂t = u +

∂2u

∂x2. (4.7)

With u(x, t) = e−az _{in (4.7) we have that}

ace−az= e−az+ a2e−az, by dividing both sides and rearranging what is left, we get

c = 1 a+ a.

This is called the dispersal relation [7, p. 442] between a and c. From this we get that the minimum value for c is 2 at a = 1.

Now, consider e−x _{and e}−ax_{for large values of x. Then for a < 1 we have}

that

e−ax> e−x

In this scenario, we have that c becomes larger as a → 0+_{. When a > 1 then}

we have that

e−ax< e−x

That is e−ax _{is bounded by e}−x _{for all x, and the leading edge is bounded by}

cmin= 2. Thus we have a wave front solution with

c = a +1

a for a < 1 and c = 2 for a ≥ 1.

The Fisher equation was used as a method to describe the spread of an ad-vantageous gene in a population. It has also been used in other areas, such as physics, astrophysics and chemistry where it can either models dispersal-diusion or heat; see [7, p. 400].

Chapter 5

Age dependent models with

spatial structure

A big area in the eld of population dynamics are models where both the age of an individual as well as where the population grows is taken to account. We shall here consider models with age and spatial structure.

5.1 An extension of the von Foerster equation

The following model was developed by Langlais [6] based upon previous models such as the von Foerster model in Chapter 3. In this model, a population is governed by the balance law (see Section2.1)

ut+ ua= −∇ · q + s, x ∈ Ω, t, a > 0, (5.1)

where u = u(x, t, a) is the population distribution, q = k∇u, for some con-stant k, is the so called random motion of the population, and s = s(t, x, a) is the supply (or birth and emigration) of individuals. We consider the set Ω ⊂ R2

a hospitable environment. The operator ∇ · (·) is the divergence with respect to the space coordinate x, that is

∇· = ∂x1+ ∂x2.

Assume also that s is dependent only on the death rate: s(x, t, a) = −µ(x, a, P (x, t))u(x, t, a), where P (x, t) = Z ∞ 0 u(x, t, a) da, (5.2)

which is the populational density at a given coordinate x and given time t. Compare s(x, t, a) to the right hand side of (3.2) and note how the equations given by Langlais and von Foerster are quite similar.

As usual we need some initial conditions. The birth process is given by u(x, t, 0) =

Z ∞

0

β(x, a, P (x, t))u(x, t, a) da, x ∈ Ω, t > 0, (5.3)

where β ≥ 0 is the birth function. Compare this to the initial condition n(t, 0) = ∞ Z 0 b(a)n(t, a) da, t > 0,

of the von Foerster equation at (3.4). The following condition is the one that that diers from the von Foerster equation as it describes what will happen on the boundary of Ω in the x coordinate, namely the Dirchelet boundary condition u(x, t, a) = 0 x ∈ ∂Ω, t > 0 (5.4) where ∂Ω is the boundary of the set Ω, which implies that the population can not survive on the boundary, in other words Ω can be seen as a hospitable environment. The nal condition is the following initial condition for t:

u(x, 0, a) = u0(x, a) ≥ 0, x ∈ Ω, t > 0.

The main equation (5.1) can now be changed into

ut+ ua− k∆u + µu = 0. (5.5)

The death and birth rate are divided into two parts respectively: β = βn(a)β(p)e

where βn(a)is the natural birth rate and β(p)eis the birth rate for the external

process, such as emigration, where p is a given population size. These functions have the following properties:

(i) 0 ≤ βn(a) ≤ ¯β < ∞, for some A0< ∞, where a > 0

(ii) supp βn ⊂ [0, A0], for some A0< ∞there exists

some 0 < r < A0such that βn> 0 on (r, A0)

(iii) βe(0) = 1, βe0 ≥ 0, βe(p) → ∞, as p → ∞.

Here supp denotes the support of βn(a)on [0, A0], which we dene as

supp(βn) = {a ≥ 0 : βn(a) 6= 0}.

Further, we let µ be divided into the sum

µ(x, a, p) = µn(a) + µe(p).

The death function µn, which represents death caused by natural reasons,

sat-ises

0 ≤ µn(a) ≤ ¯µ < ∞, a > 0.

The external death rate, µewhich takes for example overcrowding into account,

satises

5.2. Separation of variables of equation (5.5) 29

5.2 Separation of variables of equation (

5.5

)

We are going to solve Equation (5.5) by what is called separation of variables. The reader can read more on how the equations and functions that appears in this section are derived in Appendix B.3. First, let r∗ _{be the solution for the}

characteristic equation

Z ∞

0

βn(a)π(a)e−rada = 1 (5.6)

which is usually referred to as the instinct growth rate when no diusion takes place. The function π(a) in (5.6) is dened by

π(a) = exp  − Z a 0 µn(σ) dσ  , a > 0. (5.7)

Note that (5.6) is the same integral equation (in a sense) as the integral equa-tion (3.9). Similar to Section3.2we studied the long time behaviour of the von Foerster equation. There, we saw that a solution u would be of the form

u(t, a) = eγtr(a), t, a > 0.

Here, on the other hand, we shall let the solution u be of the form

u(x, t, a) = ϕ(a)P (x, t), x ∈ Ω, t, a > 0 (5.8) We shall also let ϕ(a) be normalised by

Z ∞

0

ϕ(a) da = 1.

Substituting (5.8) into (5.5) and its intial and boundary conditions, we get the PDE              ut− k∆u + ua= (ue(P (x, t)) + un(a)) u, x ∈ Ω, t > 0, a > 0 u(x, t, a) = 0, x ∈ ∂Ω, t > 0, a > 0 u(x, 0, a) = p0(x, a), x ∈ Ω, a > 0 u(x, t, 0) = βe Z ∞ 0 βn(a)u(x, t, a) da, x ∈ Ω, t > 0 . (5.9)

The following result in this section are based on the equations derived in Ap-pendixB. We have from (5.5), with substitution (5.8), that

ϕ(a) = ϕ(0) exp  −ra − Z a 0 µn(s) ds  = ϕ(0)π(a)e−ra, (5.10) where π(a) is dened as (5.7). Note that this formula is a similar result to that of (3.10) in Section 3.2. We also have that P (x, t) satises the parabolic equation      Pt− k∆P = P (r − µe(P )), x ∈ Ω, t > 0 P (x, 0) = p0(x)p, x ∈ Ω P (x, t) = 0, x ∈ ∂Ω, t > 0. . (5.11)

For both (5.10) and (5.11) we have that r is the separation constant. For more information about derivation for these equation the reader can refer to Appendix

B. The following lemma can be used to determine the existence of a separable solution.

Lemma 5.2.1. Suppose that βe(p) = 1, where p ≥ 0, and let a r∗ be a root

of the characteristic equation (5.6). Then there exists a separable solution to Equation (5.9) if and only if

Z ∞

0

π(a)e−r∗ada < ∞. (5.12)

Proof. See [6]

Note that it is sucient that r∗_{is positive for (5.12) to converge, since π(a) ≤ 1,}

and so _{Z ∞} 0 π(a)e−r∗ada ≤ Z ∞ 0 e−r∗ada = 1 r∗,

if r∗ _{is positive. Assuming that a solution for (5.5) is bounded over Ω × (0, 1).}

The ω-limit

ω(p0) = {R(x) ∈ H01(Ω) ∩ C( ¯Ω) : ∃tn→ ∞such that

P (·, tn) → R(·) uniformly on Ω as n → ∞},

where H1

0(Ω) is the Sobolev space (see AppendixC.3), is non-empty [3],

con-nected and consists of nonnegative solutions of the following ODE [6]: (

−k∆R(x) = R(x) (r∗− µe(R)) , x ∈ Ω

R(x) = 0, x ∈ ∂Ω. (5.13)

Further, we conclude the following theorems, but rst we shall introduce a new denition.

Denition 5.2.1 (Eigenvalue and eigenfunction of the Laplace operator). Let λ be a scalar and let w 6= 0 be a function. Then if

(

∆w = λw, on Ω w = 0, on ∂Ω,

we say that w is an eigenfunction and λ is the corresponding eigenvalue. We will also refer to this property by saying (λ, w) is an eigenpair.

In the two following theorems, we denote λ1 as being the rst eigenvalue of

the Dirchelet problem for the −∆ operator [6]: (

−∆w(x) = λw(x), x ∈ Ω w(x) = 0, x ∈ ∂Ω.

Remark 5.2.2. All eigenvalues for the Dirchelet problem for the −∆ operator are in fact real and positive. The eigenvalues can be order in following way:

0 < λ1≤ λ2≤ . . . .

We usually refer the smallest eigenvalue as λ1.

Theorem 5.2.2. Put m = inf{µe(p), p ≥ 0} and assume that the assumptions

in this chapter holds. Then if r∗_{< kλ} 1+ m

u(·, t, ·) → 0 uniformly on Ω × [0, ∞) as t → ∞.

5.3. Solution of equation (5.14) 31

Proof. See [12]

Recall our assumptions for µe in Section 5.1, which states that µe(p) is

increasing and has it smallest value µ(0) = 0. Then Theorem 5.2.2 implies that m = 0, and so r∗_{= kλ}

1.

For larger r∗_{we have the following theorem.}

Theorem 5.2.3. Let r∗ _{> kλ}

1 and suppose that there exists a p∗ > 0 such

that µe(p∗) = r∗. Then if u(x, t, a) is a separable solution of (5.5),

u(x, t, a) → ϕ(a)R(x), as t → ∞, uniformly on Ω × [0, ∞). Proof. See [6]

We have now reduced our equation to a new PDE for the long-term behaviour of a solution u(x, t, a). We will know get the long-term solution, v(x, a) ≥ 0 (instead of u(x, t, a)), which is the population density over Ω × [0, ∞). The PDE is the following:

                 va− k∆v = −(µn(a) + µe(R))v, x ∈ Ω, a > 0 v(x, 0) = βe(R(x)) Z ∞ 0 βn(a)v(x, a) da, x ∈ Ω v(x, a) = 0, x ∈ ∂Ω, a > 0 R(x) = Z ∞ 0 v(x, a) da, x ∈ Ω . (5.14)

5.3 Solution of equation (

5.14

)

The solution for (5.14) is given by the following theorem.

Theorem 5.3.1. Assume that all assumptions so far in Chapter 5hold. Any steady state solution of (5.14) is separable, that is, a steady state solution is of the form v(x, a) = ϕ(a)R(x), with ϕ(a) as in (5.10) and R(x) as in (5.13), and r = r∗ _{as the root of the characteristic equation}

Z ∞

0

βn(a)π(a)e−rada = 1.

Proof. Let v(x, a) be a given separable solution for (5.14). Then we separate the solution v(x, a) = ϕ(a)R(x), which results in the two equations

ϕ0+ µn(a)ϕ(a) = rϕ(a)

with initial condition

ϕ(0) = Z ∞ 0 βn(a)φ(a) da and −k∆R0(x) + µe(R(x))R(x) = rR(x)

with initial condition

where r is the separation constant. Let (λj, wj), j ≥ 1, be eigenpairs for the

Dirichlet opertor

−k∆ + µe(R(x))

Then we have that, by denition, (

−k∆wj+ µe(R(x))wj= λjwj, in Ω

wj= 0, on ∂Ω

. (5.15)

We can write a solution as a linear combination of all eigenfunctions. Since H1 0(Ω)

is a Hilbert space, this linear combination will converge. Let v(x, a) =X

j≥1

ϕj(a)wj(x)

where ϕj(a)is a eigenfunction for the _dad + µn(a)operator [1, p. 331]. Then we

have from the equation for ϕ(a), that every ϕj(a)satises

ϕ0_j+ µn(a)ϕj(a) = λjϕj(a)

which implies that

ϕj(a) = ϕj(0)π(a)e−λja.

Then we can extend our solution as v(x, a) =X j≥1 ϕj(a)wj(x) = X j≥1 ϕj(0)π(a)e−λjawj(x). Using that v(x, 0) = Z ∞ 0 βn(a)v(x, a) da we obtain that v(x, 0) =X j≥1 ϕj(0)π(0)e−λj0wj(x) =X j≥1 ϕj(0)wj(x) = Z ∞ 0 βn(a)v(x, a) da = Z ∞ 0 βn(a) X j≥1 ϕj(0)π(a)e−λjawj(x) da =X j≥1 ϕj(0)wj(x) Z ∞ 0 βn(a)π(a)e−λjada.

Especially, it is true that if X j≥1 ϕj(0)wj(x) = X j≥1 ϕj(0)wj(x) Z ∞ 0 βn(a)e−λjada then either ϕj(0) = 0 or R ∞ 0 βn(a)π(a)e λja_{da = 1.}

Since the characteristic equation (5.6) as an unique root, then there is for exactly one m ≥ 1 such that λm satises (5.6). Then we have that for all i ≥ 1,

except i = m, ϕj = 0. Thus

5.4. Some nal remarks 33

5.4 Some nal remarks

We have now shown that Equation (5.14), along with its boundary conditions, has an asymptotic solution for large t. This was shown by using the method of separation, where the solution u(x, t, a) was split into two functions, such that the age variable, a, was separated from the space and time variables x and t. In other words,

u(x, t, a) = ϕ(a)P (x, t).

Then, by Theorem 5.2.2 and Theorem 5.2.3, we saw that P (x, t) will tend to R(x) as t → ∞, and concluded further that the long-term solution

v(x, a) = ϕ(a)R(x),

as the limit of u(x, t, a) as t → ∞. Finally, the last theorem concluded that there is an unique and explicit solution for v(x, a).

Appendix A

Phase plane diagram and

stability

In Section4.1 we studied the stability of the travelling wave solution u(x, t) = f (z), where z = x − ct for the equation (4.7), where c ≥ 0. In the following section, we will derive equations for the eigenvalues of this ODE. Thus determine the stability of the equilibrium points.

We shall rst consider some theory. Let the following equation be a linear autonomous system, be of the form

f0 g0  =a b c d  f g  + C (A.1)

for functions f(z) and g(z), and C is a constant vector. Let (f, g) = (f0, g0)

be an equilibrium point for this system. We can deduce the stability of this equilibrium by the following theorem.

Theorem A.1. Let

x0(t) = Ax

be a linear autonmous system of dierntial equations os the second order (dim A = 2 × 2). Then the stablity of the equailibrium point is given by the the sign of its eigenvalues and whether they have complex components. In particular, if the eigenvalues are

both postive and real unstable

both neqative and real stable

both with a nonzero imaginary part spiral node one with a nonzero imaginary part saddle node

Further if the real parts are both positive and negative, then the equilibrium is undecided whether its a stable or unstable.

Proof. See [4, pp. 488-500]

However, it is hard to conclude the eigenvalues if the system of equations is non-linear. To deal with this problem, consider the autonomous system

f0 g0  =U (f, g) V (f, g)  (A.2) Karlsson, 2017. 35

where U(f, g) and V (f, g) are C1 _{functions. Then we have}

(

U (f, g) ≈ U (f0, g0) + (f − f0)∂U_∂f + (g − g0)∂U_∂g

V (f, g) ≈ V (f0, g0) + (f − f0)∂U_∂f + (g − g0)∂U_∂g

(A.3) but since U(f0, g0) = V (f0, g0) = 0we have that (A.3) becomes

(

U (f, g) ≈ (f − f0)∂U_∂f + (g − g0)∂U_∂g

V (f, g) ≈ (f − f0)∂U_∂f + (g − g0)∂U_∂g.

(A.4) Thus we have reduced the system. In particular, by doing the procedure above (4.7) have been reduced to a linear system, whose eigenvalues are easier to nd. This technique is called linearizion.

Recall the sher equation Fisher equation in the form of u = f,

f00+ cf0+ f (1 − f ) = 0 (A.5)

We had the phase plane trajectories denoted by dg

df =

−cg − f (1 − f )

g (A.6)

where g = f0_{. Then (f, g) = (0, 0) and (1, 0) are equilibrium points However, to}

decide the stability of these points we have to use a dierent approach to that in Section1.1. See (A.6) as a system of two equations, namely

( f0= g

g0= −cg − f (1 − f ) (A.7)

or expressed in matrix form f0 g0  =U (f, g) V (f, g)  (A.8) Here by using the formula we found from (A.4) we have that

∂U ∂f ∂U ∂g ∂V ∂f ∂V ∂g ! = 0 1 −c −1 + 2f  . So we nd that for the equilibrium (f, g) = (0, 0),

f0 g0  = 0 1 −1 −c  (A.9) and at the other equilibrium (f, g) = (1, 0) that

f0 g0  =0 1 1 −c  f g  (A.10) The eigenvalues for the 2 by 2-matrices determined the stability of each equi-librium point. Thus, with some linear algebra, we nd that the eigenvalues for (A.9) are

λ = −1 2



37

and for the eigenvalues for (A.10) are

λ = −1 2



c ± (c2+ 4)1/2. (A.12)

According to Theorem A.1 we have that the equilibrium point (f, g) = (0, 0) is unstable and the other equilibrium point (f, g) = (1, 0) is neither stable or unstable.

In FigureA.1below there are some examples of dierent equilibrium points: unstable, stable, and one that has neither of the properties.

0 0

(a) The phase portrait for a system with an unstable equilibrium point

0 0

(b) The phase portrait for a system with an stable equilibrium point

0 0

(c) The phase portrait for a system with that has neither a stable nor a unstable equilibrium.

Appendix B

Method of separation for

Langlais equation

In Chapter 5, we performed the the technique called separation of variables [10, p. 141], on the equation              ut− k∆u + ua= (ue(P (x, t)) + un(a)) u, x ∈ Ω, t > 0, a > 0 u(x, t, a) = 0, x ∈ ∂Ω, t > 0, a > 0 u(x, 0, a) = p0(x, a), x ∈ Ω, a > 0 u(x, t, 0) = βe Z ∞ 0 βn(a)u(x, t, a) da, x ∈ Ω, t > 0 . (B.1)

We shall assume that u(x, t, a) = ϕ(a)P (x, t), that is, u is can be separated into two parts where one function ϕ(a) and P (x, t). This is known as an ansatz [10, p. 141]. This approach is only really motivated with that it sometimes work. For more lengthy discussion on the method of separation, see [11].

However, the main equation of (B.1) can be written as ua− µn(a)u = −ut+ k∆u + µe(P (x, t)).

With u = ϕ(a)P (x, t) we have

ϕ0P − µn(a)ϕP = −ϕPt+ kϕ∆P + µe(P (x, t))ϕP,

which after dividing both sides with ϕP 1 ϕϕ 0_{− µ} n(a) = − 1 PPt+ k 1 P∆P + µe(P (x, t)) = r. (B.2) Now the age variable a has been separated from the space and time variables (x, t). Here we let r be the relation constant between ϕ(a) and P (x, t). Thus the left hand side of (B.2) becomes an ODE:

1 ϕϕ 0_{− µ} n(a) = r, which gives ϕ0− µn(a)ϕ − rϕ = 0, (B.3) Karlsson, 2017. 39

and the right hand side becomes a PDE: −1

PPt+ k 1

P∆P + µe(P (x, t)) = r

which can be rearranged as

−Pt+ k∆P + µe(P (x, t))P − rP = 0.

With the initial condition a = 0, (B.3) becomes    ϕ0− µn(a)ϕ − rϕ = 0 a > 0 ϕ(0) = Z ∞ 0 βn(a)ϕ(a) da a = 0 ,

which we can solve. With the integrating factor − Z a 0 (µn(s) + r) ds = −ra − Z a 0 µn(s)ds

we obtain the solution

ϕ(a) = ϕ(0) exp  −ra − Z a 0 µn(s) ds  .

For the equation governing P (x, t), we are at the moment satised with its current form      µe(P (x, t))P − rP = Pt− k∆P, x ∈ Ω, t > 0 P (x, t) = 0, x ∈ ∂Ω, t > 0 P (x, 0) = p0(x, a), x ∈ Ω. ,

Further analysis is covered in Chapter 5, where we see that a solution P (x, t) tend to R(x) as t → ∞.

Appendix C

Functional Analysis

For the later part of the thesis, we need some understanding of functional anal-ysis, especially how we measure the size a function and dene how we can say that functions are orthogonal to each other. Our goal with this section is to dene Sobolev spaces, especially H1

0 where a solution u of a PDE will be an

element of this space. First we dene normed spaces.

C.1 Normed spaces

We start of with an important denition.

Denition C.1.1 (Metric). Let x, y and z be elements of a set X. We say that the pair (x, d) is a metric space, where d : X × X → R is a metric which has the following properties:

(i) d(x, y) ≥ 0

(ii) d(x, y) = 0 if and only if x = y (iii) d(x, y) = d(y, x)

(iv) d(x, y) ≤ d(x, z) + d(z, y) (triangle inequality)

Remark C.1.2. X does not necessarily have to be a vector space, but for the purposes of this thesis, that case will not be covered.

Denition C.1.3 (Norm). Let x, y and z be elements of a vector space X. A norm || · || satises

(i) || x ||≥ 0

(ii) || x ||= 0 if and only if x = 0 (iii) || x + y ||≥ || x ||+|| y || (iv) || αx ||= |α| || x || α ∈ C.

(C.1) Remark C.1.4. We can dene a metric through a norm by

d(x, y) = || x − y ||.

Denition C.1.5 (Normed Space). A vector space X is called a normed space if it can be equipped with a norm || · ||, which we can also denote as (X, || · ||).

C.2 Inner product spaces

In this part, we shall dene what is called inner products. With an inner product it is possible to dene whether two elements of a vector space are orthogonal to ea.

Denition C.2.1 (Inner product). Let x, y and z be elements of a vector space X. A inner product h·, ·i satises

(i) hx, xi ≥ 0

(ii) hx, xi = 0 if and only if x = 0 (iii) hx + y, zi = hx, zi + hy, zi (iv) hαx, yi = αhx, yi α ∈ C

(v) hx, yi = hy, xi.

Example C.2.2. The dot product from linear algebra is an inner product. Similar to normed spaces we dene inner product spaces by the following denition

Denition C.2.3 (Inner product space). A vector space X equipped with the inner product h·, ·i is called an inner product space. This is denoted by (X, h·, ·i). Remark C.2.4. An inner space is also a normed space. This can be seen by dening a norm by

|| x ||=phx, xi.

Before moving on to Sobolev spaces in the next section, we will rst dene a two important spaces in the following denitions.

Denition C.2.5 (The space Lp_{). Let X be the vector space of all continuous}

real valued functions dened on the interval [a, b] with a norm dened as || x ||Lp= Z b a xp(t) dt !1/p . An element x ∈ X is also an element in Lp_{[a, b]if,}

|| x ||Lp< ∞.

We also dene Lp _{as L}p

(R).

Denition C.2.6 (Compactness). In a nite dimensional normed space X, any subset M ⊂ X is compact if and only if M is closed and bounded.

Denition C.2.7 (Support). Let function f(x) be a function dened on [a, b]. We dene the support of f(x), as

supp f (x) = {x ∈ [a, b] : f (x) 6= 0}. Denition C.2.8 (The space C1

c). The space C 1

c[a, b]is the vector space of all

dierentiable continuous functions dened on [a, b] with compact support. We also dene C1

C.3. Sobolev spaces 43

C.3 Sobolev spaces

In the later part of the thesis, Sobolev spaces and its properties are integral to the solution. Usually, a solution for a dierential equation is an element of a Sobolev space. The denitions and theorems throughout the remainder of this section is mainly based upon [1, pp. 201-312], but in a more compact form. Denition C.3.1 (The Sobolev space W1,p_{). Let Ω ⊆ R}n_{. The Sobolev space}

W1,p is the collection of functions u ∈ Lp(Ω) such that there exist functions gi∈ Lp(Ω), for i = 1, . . . , n, satisfying Z Ω uϕ0_i= − Z Ω giϕi,

for any ϕi ∈ Cc1(Ω), for i = 1, . . . , n. An important spacial case of a Sobolev

space is H1_{(Ω) which is given by letting p = 2.}

Sobolev spaces are normed spaces according to the following proposition. Denition C.3.2. Dene the following norm

|| u ||W1,p= || u ||_Lp+|| u0||_LP

for u ∈ W1,p_.

Proposition C.3.1. || · ||W1,p is in fact a norm.

Proof. See [1, p. 203].

We can also dene an inner product for the space H1_.

Denition C.3.3. Dene h., .iH1 by the following.

hu, viH1 = hu, vi_L2+ hu0, v0i_L2 =

Z

Ω

(uv + u0v0) for u, v ∈ H1_.

Proposition C.3.2. The inner product h., .iH1,p is in fact a norm.

Proof. See [1, p. 203].

For a PDE, there usually is some boundary value condition associated with it. The Sobolev spaces , W1,p

0 , especially H 1

0 are important for these kinds of

problems, which we will dene below. Denition C.3.4. Let 1 ≤ p < ∞; W1,p

0 (Ω) denotes the closure of Cc1(Ω) in

W1,p_{(Ω).The special case when p = 2, we have}

H₀1= W₀1,2.

An important property of these spaces are given by the following theorem. Theorem C.3.1. Let Ω ⊆ Rn_{, have C}1 _{boundary ∂Ω and let u ∈ W}1,p_{(Ω) ∩}

C(Ω). Then u ∈ W₀1,p if and only if u = 0 on ∂Ω.

Proof. Since some concepts required by the proof are not covered here. The interested reader is referred to [1, pp. 288-289].

Bibliography

[1] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Dierential Equations, Springer, London, 2011.

[2] Campbell, F.C., Elements of Metallurgy and Engineering Alloys, AMS In-ternational, Ohio, 2008.

[3] Dafermos, C., Asymptotic behavior of solutions of evolution equations. In Crandall, M.G (ed.), Nonlinear evolution equation. New York: Academic Press, 1978.

[4] Edwards, C.H., Penney, D.E, Dierential Equations and Boundary Value Problems, Pearson, New Jersey, 2008.

[5] Ellner, S.P., Guckenheimer, J, Dynamic Models in Biology, Princeton Uni-versity Press, Princeton 2011.

[6] Langlais, M., Large time behaviour in a nonlinear age-dependent popu-lation dynamics problem with spatial diusion, Journal of Mathematical Biology, 26, 319-346.

[7] Lions, P. L., On the existence of solutions of semilinear elliptic equations. SIAM Review 24, 441-468.

[8] Murray, J.D., Mathematical Biology: I. An Introduction, Third Edition, Springer-Verlag, Berlin, 2002.

[9] Ross, C., Dierential Equations: An Introduction with Mathematica, Springer, Sewanee, 2004.

[10] Strauss, W., Partial Dierential Equations: An Introduction, John Wiley & Sons, Providence, 2007.

[11] Teschl, G., Ordinary Dierential Equations and Dynamical Systems, Amer-ican Mathematical Society, Vienna, 2012.

[12] http://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx

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c

2017, Anton Karlsson