Modelling Range Expansion of Invasive Species

Kyle Stevens

Supervised by James McCoy and Ngamta

Thamwattana University of Wollongong

Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute.

Abstract

Various methods for modelling the invasion of a territory by a species were discussed by Shigesada

and Kawasaki (1997)[1]. The aim of this project was to research the methods for formulating some

of these models with an emphasis on finding the speed of expansion of an invasive species. The

particular models analysed are the reaction-diffusion equations with logistic (Fisher’s Equation)

and Lotka-Volterra source terms. Travelling wave solutions to these models are analysed with their

wave speeds being derived. The wave speed of all the solutions to the models are governed by a

central equation, c = 2√εD where ε is the growth rate of a species and D is the diffusivity of the

species. With two species present, the wave speed of the invasive species is found to be altered to

c = 2

√ε1D1

(1− ε2µ12

ε1µ11

)where the µ terms represent inter- and intraspecific competition, with ε1

being the growth rate of the invasive species, D1 is the diffusivity of the invasive species and ε2 is

the growth rate of the resident species.

2

Contents

1 General Features of Expansion 5

2 Uniform Diffusion and Biological Growth 7

2.1 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Reaction Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Calculating the Wave Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 An Example: Cane Toads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Expansion in an Heterogeneous Environment 16

3.1 Fixed Patch Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Random Patch Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Invasion of Competing Species 20

4.1 Complete Displacement of Resident Species . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Coexistence with the Resident Species . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 The Fugitive Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Summary 27

3

Introduction

Migration is a natural process for many types of organisms. There are many reasons that could

cause a species to venture from their native land to new pastures, including loss of habitat, too

high a population density or being displaced by another species to name a few.

When this movement to a new location disrupts the equilibrium of the environment to a high

enough degree, the perpetrating species is named an invasive species. Often such species are in-

troduced by human means, whether by accidental stowaways (e.g. in ships and aircraft) or an

intentional introduction to deal with some problem (e.g. cane toads, camels).

The range expansion of these invasive species is of particular interest. Knowing how long a

species will take to occupy the entirety of a given area will shape the urgency and methods for the

action taken against the species. This report will look at the methods and models discussed in [1].

Section 1 concerns the general features of invasion that every invasive species exhibits such as

the three phases of an invasion and the types of expansion.

Section 2 considers modelling a single species with reaction-diffusion equations in an homoge-

neous environment with focus on Fisher’s equation and the speed of its travelling wave solutions.

Section 3 then uses the models from section two but with a focus on a patchy environment,

specifically two types of patches, and how the wave speed is altered.

Section 4 concerns the modelling of an environment with two species, a resident and an invader,

and how this affects the speed of invasion.

Finally, the summary is given in Section 5.

4

1 General Features of Expansion

In order for an invasion to successfully occur, the very first thing a species must address is the

initial population size. A minimum population size required for invasion has been estimated to be

given by 3ln(b/d) , where b is the birth rate and d is the death rate of the species[2].

Once a species has taken root in a new land, an establishment phase with very little range

expansion often takes place. There are many theories on why this happens, such as a species

growth being subject to an Allee effect, which means there are less reproductive successes at low

population density, or the dispersal of the initial creatures happens so quickly that the species exists

below a threshold density for detection for some time, etc... .

Following the establishment phase in an expansion phase, where the species has reached a high

enough local density around their invasion point that they must garner new territory to grow their

population further. The rates of expansion depend on the species, however they all follow one

of three patterns: linear, biphasic and increasing with time. Shigesada and Nanako note that

species exhibiting linear expansion often have their offspring settling within a short distance of

their parent generation whereas the species with biphasic expansion have their offspring dispersing

a short distance away from the main colony, only to rejoin it later[1]. The species which employ

long distance dispersal of offspring and form satellite colonies are the ones with an accelerating rate

of expansion.

The last phase of invasion is the saturation phase. This occurs when a species has reached the

maximum amount of area it can attain and can no longer expand. Reasons for this include things

such as natural barriers like coastlines or mountain ranges preventing further movement, or man

made blockages like towns and cities. Plotting the range distance against time for an arbitrary

species gives a sigmoidal shape, as shown in Figure 1.

5

Figure 1: Range expansion of an arbitrary species over time, the three phases are labeled clearly and the expansionphase is of an accelerating type.

In the text studied, the vast majority of equations concerned two-dimensional equations (under

the assumption that height or depth is inconsequential) and as such involve a coordinate system

with orthogonal x and y directions, however in the case of radially symmetrical expansion (which is

ideal), the measurement of range distance can be reduced to a single variable r, representing radius.

For circular expansion, r, is calculated by rearranging A = πr2 to r =√

Aπ , where A is the area

the species occupies. If the spread is irregular, but still circular in nature, then the equation for

the area of a circular sector may be used, namely A = θ2r

2 so that r =√

2Aθ , where θ is the angle of

the segment. When an area presents itself as too irregular but boundary measurements have been

taken over time we can calculate the radius with the following: ∆r =

√∆r2min+∆r2max

2 where ∆rmin

and ∆rmax are the minimum and maximum range differences measured between two consecutive

boundary measurements Ωi and Ωi+1.

6

2 Uniform Diffusion and Biological Growth

2.1 The Diffusion Equation

Diffusion has been used to describe the movement of continuous densities (e.g. heat) through objects

and across surfaces since Fourier proposed the heat equation in 1822[3]. In [1] the populations of

invasive species were considered as a continuous densities of creatures, subject to space and time,

n(x, y, t), and therefore the diffusion equation is used to describe the movement of said populations.

∂n

∂t= D(

∂2n

∂x2+∂2n

∂y2). (1)

Equation (1) is the diffusion equation in two spatial dimensions, x and y, where D is the

coefficient of diffusion and determines how fast the density spreads out through the object or

surface. As most animals live solely on land, we can make an assumption that a depth coordinate,

z, is not needed, which serves to simplify all the diffusion models this report will cover. In the

case of radial measurements for a species, letting r =√x2 + y2, converts (1) to an equation in one

spatial variable,

∂n

∂t= D

(∂2n

∂r2+

1

r

∂n

∂r

). (2)

To solve either of these equations we need an initial condition. As species are introduced, the

density of a population is a tight, localised distribution. For t = 0, n(r, 0) = n0δ(r), where n0 is

the initial number of creatures in the invasion and δ(r) is the probability density function of the

distribution of the density of the species (e.g. Gaussian, uniform, etc...). There are also implicit

boundary conditions with r →∞, n→ 0 and ∂n∂r → 0. Solving (2) with these conditions gives the

following solution:

n(r, t) =n0

4πDte−

r2

4Dt . (3)

7

This solution looks very similar to the probability density function of the Gaussian distribution,

however as t → ∞, the density goes to zero everywhere. As this does not happen to species, the

diffusion equation alone is not enough to model the range expansion of an invasion; there must be

a source term.

2.2 Population Dynamics

In [1], two types of population growth were considered for a single species: Malthusian growth

and logistic growth. Malthusian growth is best described as exponential growth whereas logistic

growth, when plotted, has a more sigmoidal curve. The equations for these types of growth are

given respectively by

dn

dt= εn, (4)

dn

dt= (ε− µn)n, (5)

with ε being a growth constant and µ being a density based loss constant. Solving (5) shows

that when t → ∞, the population density has an equilibrium point at the carrying capacity, εµ .

The carrying capacity of a species is defined as the maximum population size (in this case density)

of the species the environment can sustain indefinitely. The Malthusian growth model has no such

carrying capacity and shows unbounded growth as t → ∞. These equilibria are evident in their

analytic solutions;

n(t) = n0eεt, (6)

n(t) =n0

εµeεt

εµ

+ n0(eεt − 1). (7)

8

2.3 Reaction Diffusion Models

Combining either of these population models with the diffusion equation yields the two models

primarily discussed in [1]. The first and simpler model is known as Skellam’s model, named after

John Skellam who investigated the problem as a potential population distribution model in 1951[4].

The solution is extremely similar to the solution of the diffusion equation,

∂n

∂t= D

(∂2n

∂x2+∂2n

∂y2

)+ εn, (8)

n(x, y, t) =n0

4πDteεt−

x2+y2

4Dt . (9)

Figure 2: Solutions to Skellam’s model with D = 1 and ε = 1 over a small time period. The first three time stepsgive good approximations to a real life species but the unbounded nature of the model takes over quickly

The εt in the exponential term in (9) leads to the model giving unbounded population density

values as time goes to infinity, thus it is a poor equation for population modelling over large

timescales. However, for a species that has not yet hit their carrying capacity and also has a small

growth rate, Skellam’s model is capable of giving accurate predictions. An example of a species

9

that would be easy to model with Skellam’s model is the feral camel. Introduced to the Australian

outback in the mid 1800’s to aid exploration, the camels proved useful until the advent of motor

vehicles which left many camels abandoned. Subsequently these camels thrived in the deserts of

Western Australia and feral populations emerged, which have since grown to numbers of up to

600,000 camels across the outback[5]. As the feral camel has a long gestation period of 15 months

and gives birth to a single child at a time, the species has a low intrinsic growth rate. Coupled with

the vast expanses of land available to the camels still, the species is still considered to be growing

exponentially and would be appropriately modeled by Skellam’s model.

The second single-species model studied is the well known Fisher’s equation, named after Ronald

Fisher who proposed the equation be used in the context of biological modelling in 1937[6]. This

model has more use over large timescales due to the nature of an in built carrying capacity, εµ , which

is the same value for that of the logistic growth model. As most species tend to have a maximum

population density in a given area, their range expansion behaviours can be modeled well using

Fisher’s equation,

∂n

∂t= D

(∂2n

∂x2+∂2n

∂y2

)+ (ε− µn)n. (10)

An analytical solution to this equation proved to be beyond the scope of this report, thus

numerical methods were employed via Maple to yield solutions and plots. Fisher’s equation can

also be reduced to a one spatial dimension form using a radius r just like the diffusion equation

was above. This reduction has the form:

∂n

∂t= D

(∂2n

∂r2+

1

r

∂n

∂r

)+ (ε− µn)n. (11)

This form is much more useful to analyse the travelling wave solutions to this model. A travelling

wave solution is a solution of the form U(z) with z = r − ct where c is the wave speed. As the

radius can only be positive, only a right travelling wave solution is considered, hence z = r + ct

was not examined in the text. U(z) also has two implicit boundary conditions as seen from Figure

10

Figure 3: Travelling wave solutions to Fisher’s equation. D = 1, ε = 1 and µ = 1.

3, U(−∞) = εµ , U(∞) = 0. The use of the travelling wave solution is that it matches the real life

phenomena of the species having a high density in areas inside the range front (to the left in one

dimension), having low densities near the range front and having zero density outside the range

front. From Figure 3 it is plain to see that these travelling wave solutions exist as trajectories from

the carrying capacity equilibrium point of the model to the zero equilibrium point of the model.

2.4 Calculating the Wave Speed

The speed at which these waves propagate can thus be found by analysing the ODE system which

arises by substituting U(z) = n(r, t). It should be noted that for the following calculations, large r

is assumed due to analysing the equilibrium points, so the 1r∂n∂r term in (11) is assumed negligible.

∂U

∂t=dU

dz

dz

dt= −cdU

dz,

∂2U

∂r2=d2U

dz2,

11

−cdUdz

= Dd2U

dz2+ (ε− µU)U. (12)

By letting dUdz = V , equation (12) becomes a system of two ODEs:

dU

dz= V,

dV

dz= − c

DV − (ε− µU)

DU.

(13)

This system has two equilibrium points of the form (U, V ), at z → −∞ and z →∞ and they are

( εµ , 0) and (0, 0) respectively. The travelling wave solution of the system is known as a heteroclinic

orbit. A heteroclinic orbit is one which connects two equilibrium points together in a phase plane.

In order for this to occur, one of the equilibrium points must be unstable, potentially a saddle point,

and the other must be an attracting node. The stability of these equilibrium points were analysed

via their eigenvalues and also a phase portrait. Figure 4 shows that the(εµ , 0)

equilibrium point

is a saddle point and that (0, 0) is an attracting node however to prove this the eigenvalues must

be calculated.

Figure 4: A phase portrait of the U-V plane with trajectories coloured in green and the red arrows denoting directionfields of the trajectories. The equilibrium points are the blue circles.

12

In order to calculate the eigenvalues, the Jacobian of the system must be found. The Jacobian

of a system is a matrix where the (i, j)th entry is defined as Ji,j = ∂fi∂xj

. In the case of this system,

the f ′is are dUdz and dV

dz and the x′js are U and V . Hence the Jacobian is written as

J(U, V ) =

0 1

µUD− ε−µU

D−cD

. (14)

Substituting the values of Uand V at the equilibrium points into the Jacobian gives the matrix

A from the equation Ax = λx, where λ is the eigenvalues. Rearranging this equation gives a matrix

A− λI (I being the identity matrix) whose determinant can be solved to give the eigenvalues.

A(ε

µ, 0) =

0 1

εD

−cD

,A(0, 0) =

0 1

−εD

−cD

.(15)

The matrices give the following eigenvalues respectively:

λ± =−c±

√4εD + c2

2D,

λ± =−c±

√−4εD + c2

2D.

(16)

Whilst very similar, the difference in the sign of the 4εD under the square root makes a large

difference. First examining the eigenvalues for ( εµ , 0), as√c2 + 4εD > c, the signs of λ+ and λ−

are always opposite, also they are both real. When this occurs, the equilibrium point is called an

unstable saddle point. For the (0, 0) point,√c2 − 4εD < c, which means that both the eigenvalues

are always negative. This means that the equilibrium point can either be a regular attracting node

or an attracting spiral node; either case depending if the eigenvalues are real or complex respectively.

However, due to the condition U ≥ 0 as it is the population density (which can never be negative)

13

a spiral node is impossible as a trajectory spiraling around (0, 0) would eventually have a negative

U value at some point. This means that the term√c2 − 4εD must always be positive, leading to

an inequality for c,

c ≥ 2√εD. (17)

This result means that there is an infinite number of heteroclinic orbits with a wave speed above

2√εD, however, it can be proved that if an initial distribution is concentrated within a localised

region, much like an invasive species would be, then the distribution will ultimately converge to the

minimum wave speed

c = 2√εD, (18)

providing it follows Fisher’s equation[7]. With this information, it becomes possible to achieve

good estimates for the speed at which a species range front will expand; only needing to know the

intrinsic growth rate and the diffusivity of the species, both of which can be found by experimental

data.

2.5 An Example: Cane Toads

A study in North-Western Queensland found that between the years 1980 and 1984, a population of

cane toads were found to be expanding their territory at a rates of 19−35km/year[8]. Assuming that

the region in which the cane toads were invading is homogeneous, I am able to use (18) to estimate

the rate of expansion of the cane toads in this area. The growth rate can be roughly estimated by

subtracting the death rate of the species from the birth rate, giving ε = 20− 80individuals/year. The

14

diffusivity of the cane toad can be estimated from mark-recapture data[9]using the equation

D =

([n=1]N

∑(distance from site of recapture)

total number of recaptured animals

)2

π × (time from release),

which gave D ≈ 3.2km2/year. Using these estimated variables, an estimate for c is found to be

c = 16− 32km/year, which is quite close to the true value expansion rates above.

15

3 Expansion in an Heterogeneous Environment

In the above section, the area in which a species is expanding was considered homogeneous. This

is not true to life, especially when considering an invasion over a large enough area with different

biomes. Models that take this heterogeneity into account are often written as patch models. A

patch model for Fisher’s equation is one where the diffusion coefficient and the growth constant are

variable functions in r. The simplest way of expressing this is with periodic step functions where

the area in which the species is invading is divided into evenly spaced patches of land, alternating

between favourable and unfavourable terrain. In this context, favourable terrain is land in which the

species is able to thrive and often has a higher growth rate when compared to being in unfavourable

terrain. The diffusion coefficient is not necessarily higher in favourable terrain however, as a species

could slow down in terrain it prefers and not wish to disperse as much.

In the one dimensional case with only two types of patches, these constants take the form

D(x) =

d1 x2m ≤ x ≤ x2m+1

d2 x2m+1 ≤ x ≤ x2m+2

ε(x) =

ε1 x2m ≤ x ≤ x2m+1

ε2 x2m+1 ≤ x ≤ x2m+2,

(19)

where m = 0, ±1, ±2, . . . with x0 = 0, xi+1 = xi+ li+1 and li is the width of the ith patch. Odd

numbered patches will be considered favourable, even will be considered unfavourable. Only two

types of patches will be considered, though extension to n patches is possible. Regularly arranged

patches can be thought of and solved as periodic step functions with the coefficients switching every

li units (meters, kilometers, etc. . .)

∂n

∂t= D(x)

∂2n

∂x2+ (ε(x)− µn)n. (20)

16

3.1 Fixed Patch Lengths

A method for achieving patches with fixed sizes can be achieved using a truncation-based formula

within the step function, where l1 is the size of the favourable patch and l2 is the size of the

unfavourable patch.

D(x), ε(x) =

d1, ε1 x− (l1 + l2)trunc

(x

l1+l2

)≤ l1

d2, ε2 otherwise,

(21)

where the trunc function truncates whatever value it is given.

Figure 5: Solutions of equation (23) with l1 = 2, l2 = 3, d1 = 1, d2 = 0.5, ε1 = 1, ε2 = −0.5.

Plotting the results of these fixed patch sizes leads to travelling periodic wave solutions. A

travelling periodic wave is similar to the travelling frontal waves discussed in Section 2, but with

a slight difference. Rather than there being a plateau at the carrying capacity, εµ , there is instead

a periodic wiggle which denotes the fluctuating carrying capacities between patches of land. This

can be seen in Figure 5.

As there are multiple growth rates and diffusion constants, there are different speeds at which

17

the wave front travels through patches. Defining l∗ = l1 + l2 as the spatial period, which is the

length which the distribution travels before repeating its cycle, then t∗ can be defined as the time

taken for the range front to traverse l∗. The speed at which the range front travels can then be

represented as:

c =l∗

t∗. (22)

This is still a constant speed, so a species propagating through a regularly patched landscape

would still exhibit constant expansion. The problem lies with arbitrarily small patch lengths; as

l∗ → 0, t∗ → 0 which is not taken into account by equation (21). This issue was solved and it

can be shown that the wave speed can be approximated by the following equation when the patch

lengths are sufficiently small[10]:

c =

2√〈ε〉a 〈d〉h 〈ε〉a ≥ 0

0 〈ε〉a < 0,

(23)

where 〈ε〉a and 〈d〉h are the arithmetic and harmonic means defined as:

〈ε〉a =ε1l1 + ε2l2l1 + l2

,

〈d〉h =l1 + l2l1d1

+ l2d2

.

Equation (22) has the same form as the speed derived from Fisher’s equation in an homogeneous

environment, which means that when the patch lengths become sufficiently small, a species in such

an environment would behave like one in an homogeneous environment.

3.2 Random Patch Lengths

Instead of fixing the patch lengths, allow them to be chosen from the following intervals:

18

l1 − σ1 ≤ l2m+1 ≤ l1 + σ1,

l2 − σ2 ≤ l2m+2 ≤ l2 + σ2,(24)

where l1 and l2 are the mean lengths of the favourable and unfavourable patches, and σ1 and σ2

are their respective deviations. Using lengths chosen from (23) in equation (19) gives two possible

solutions, one of which is the trivial solution with the species going extinct. When a species under

these conditions does not go extinct, however, they evolve into a travelling irregular wave. Much

like the travelling periodic wave, the species fluctuates in density along the range it inhabits, but

this time there is no fixed pattern in time or space, which would be expected from randomised

patch widths. As it turns out, the wave speed of the travelling irregular wave barely deviates from

the speed of the corresponding travelling periodic wave (when σ1 = σ2 = 0)[11]. Also similarly to

the periodic travelling wave, when the irregular patch lengths are all sufficiently small, the speed

can be calculated as:

c =

2√〈ε〉a 〈d〉h 〈ε〉a ≥ 0

0 〈ε〉a < 0,

(25)

with

〈ε〉a =ε1l1 + ε2l2l1 + l2

,

〈d〉h =l1 + l2l1d1

+ l2d2

.

So, despite the local fluctuations in the irregular case, the overall wave speed is still constant

and therefore a species propagating within an environment such as this would still exhibit linear

expansion.

19

4 Invasion of Competing Species

This section explores the invasion of a species with the presence of another species. Three cases

arise when considering a two species model where the invading species persists and successfully

invades:

1. The invading species takes over the resident species habitat via direct, aggressive competition.

2. The competition with the resident species is not as aggressive as in case 1 and the two species

end up coexisting.

3. The invading species is competitively weak but persists through superior diffusion by moving

into open spaces that arise.

Case 1 will occur when the invading species spreads its range by displacing the resident species

entirely whereas in case 3, the invading species is actively displaced by the resident species. Con-

sequently, a species following case 3 will move from open space to open space constantly; such

species are called fugitive species. The following one-dimensional two-species model describes the

interaction between two species:

∂n1

∂t= D1

∂2n1

∂x2+ (ε1 − µ11n1 − µ12n2)n1,

∂n2

∂t= D2

∂2n2

∂x2+ (ε2 − µ21n1 − µ22n2)n2,

(26)

where n1, n2 are the densities of the resident and invading species respectively, D1, D2 are

their diffusion coefficients, ε1, ε2 are their growth rates and µii, µij are the intra- and interspecies

competitions. This is a system of reaction-diffusion PDEs where the source terms come from the

Lotka-Volterra predator-prey model,

dn1

dt= (ε1 − µ11n1 − µ12n2)n1,

dn2

dt= (ε2 − µ21n1 − µ22n2)n2.

(27)

This model is equivalent to (28) when D1 = D2 = 0, i.e. the two species do not leave the area

20

they reside in. As this is a system of ODEs, phase plane analysis is very useful here. Setting the

differential equations to 0 gives the nullclines:

(ε1 − µ11n1 − µ12n2)n1 = 0⇒ n1 = 0 or n2 = ε1−µ11n1

µ12

(ε2 − µ21n1 − µ22n2)n2 = 0⇒ n2 = 0 or n1 = ε2−µ21n1

µ22

From these nullclines, which are the set of points in the phase plane where the DEs are zero,

there are four possible equilibrium points, denoted Ei = (n1, n2) :

E0 = (0, 0)

E1 =

(ε2

µ21

, 0

)E2 =

(0,ε1

µ12

)E3 =

(ε1µ22 − ε2µ12

µ11µ22 − µ12µ21

,ε2µ11 − ε1µ21

µ11µ22 − µ12µ21

)(28)

Stability analysis of these four points gives four different circumstances:

1. When ε1µ11

< ε2µ21

and ε2µ22

> ε1µ12

, species 2 will win.

2. When ε1µ11

< ε2µ21

and ε2µ22

< ε1µ12

, both species will coexist.

3. When ε1µ11

> ε2µ21

and ε2µ22

< ε1µ12

, species 1 will win.

4. When ε1µ11

> ε2µ21

and ε2µ22

> ε1µ12

, either species can win depending on the initial conditions

(the coexistence exists as an equilibrium point in this case but is unstable).

As species 2 in this case is the invading species, we are interested mainly in situations stemming

from perturbing the E1 equilibrium, that is, a situation where species 1 is already dominant. This

means that in order for the invading species to be successful, only the first two circumstances need

to be considered.

21

4.1 Complete Displacement of Resident Species

The first assumption to be made in this case is that the entire area is occupied by the resident

species such that it has reached its carrying capacity, i.e. n1(x, 0) = ε1µ11

at all points. Then an

injection of a few individuals of the invasive species occurs, which disturbs this equilibrium. In

order for the invasive species to succeed, the current steady state must be unstable to perturbation;

the conditions for this to occur are the same as those in the Lotka-Volterra model. In the case that

the resident species is displaced entirely, the spread of the invasive species can be described by a

travelling wave of constant speed. The speed of this wave can be derived by assuming that at the

wave front, n1 ≈ ε1µ11

and n2 ≈ 0, which gives the following equation:

∂n2

∂t= D2

∂2n2

∂x2+

(ε2 −

ε1µ21

µ11

)n2. (29)

This arises from the n22 term becoming negligible as n2 is already close to zero. Equation (28)

is of the same form as Skellam’s model seen in Section 2 with D = D2 and ε = ε2 − ε1µ21

µ11. Since

the wave speed in Skellam’s model is given by c = 2√εD as in Fisher’s equation, the wave speed

for the invasive species is given by:

c2 = 2

√ε2D2

(1− ε1µ21

ε2µ11

). (30)

Comparing this speed with that from Fisher’s equation, which arises from (25) when n1 = 0 (no

resident species), it follows that the ratio of the two speeds gives the effect that the resident species

has on the invasive species expansion rate,

c2

c2

=

√1− ε1µ21

ε2µ11

≡ γ < 1. (31)

This constant, γ, represents the factor by which a resident species affects the invading species

expansion speed, the square root ensures that the resident species cannot actually accelerate the

invading species invasion, which would be an interesting event were it to occur. In order for γ to

22

be as close to 1 as possible, species two must not only be better competitively (low µ21) and have

a higher growth rate (high ε2) but it must also be lucky enough such that species one has high

intraspecies competition (high µ11).

Figure 6: Solutions to the two species model with the blue curves representing the invading species and the redrepresenting the resident species. As time goes on, the red curve tends to zero as the blue curve tends to its carryingcapacity.

4.2 Coexistence with the Resident Species

For coexistence to occur, the same initial situation is considered where the resident species com-

pletely occupies the space and the invading species expansion can be described by a travelling wave.

The biggest difference between this and the total displacement is that the intraspecies competition

of species two outweighs the effect that interspecies competition has on species one. This coex-

istence case has a steady state solution where both species exist at a modified carrying capacity,

as seen in E3 from (27) and also in Figure 7. As Figures 6 and 7 appear to have a lot going on,

Figure 8 provides a snapshot in time of both models side by side, to better visualise exactly what

the difference is between the two models.

23

Figure 7: Solutions to the two species model under a coexistence scenario. The blue line represents the invadingspecies and the red the resident, both densities exhibit a nonzero carrying capacity, but also a carrying capacity lessthan their ideal.

Figure 8: Side by side comparison of the coexisting species solution and the displacement of resident species solution.Both have the blue curves as the invading species and the red curves as the resident. Both plots are at a single timet.

24

4.3 The Fugitive Species

An invasive species will become a fugitive species when it is competing for open space with another

species, but it is competitively weaker that this other species. Therefore the only way for the

invading species to survive and prosper is to diffuse much faster than the species it is competing

with. Modelling this competition for open space can be done still with (25) however the conditions

for which species wins will be fulfilled for species one. That is, ε1µ11

> ε2µ21

and ε2µ22

< ε1µ12

; but we

will now consider D1 < D2.

As the assumption that one species inhabits the entire range no longer holds, the first, more

competitive species, is considered to be the first invader, with no second species present, This

reduces (25) to Fisher’s equation, which was shown in section 3 to have a travelling wave solution

with wave speed :

n1(x− c1t), c1 = 2√ε1D1 . (32)

The second species is then introduced at what will be named time zero at a distance L units from

the origin within the wave front of the first species, with N0 individuals. This gives the following

initial condition to be applied to (25):

n1(x, 0) = n1(x),

n2(x, 0) = N0δ(x+ L).(33)

Solving (25) with the conditions in (32) gives two possible situations: The first being one where

species two goes extinct and the second being the solution where species two overtakes species one

by superior diffusive capabilities and proceeds to expand ahead of species one. By analysing the

stability of the extinction steady state, it was found that if ε1D1 < ε2D2, then the second invading

species will catch up to and pass the first species[1].

As was shown in Section 4.1, the speed of a species in competition under model (25) is given

25

by

c1 = 2

√ε1D1

(1− ε2µ12

ε1µ11

)< c1.

This means that the speed at which the first species will displace the second as it lags behind

it is c1. It should be noted that from the above condition, it is implied that c1 < c2, meaning that

when the second species overtakes the first, it will continue to widen the gap between their two

range fronts, providing that there is a continuous supply of open space.

26

5 Summary

Whilst the most crucial time period to stop an invasion would be during the establishment phase,

this is not always possible. Because of this, modelling of the expansion phase of a species is vital

to planning action on stopping an invasion before the saturation phase occurs. The speed at which

the range front of a species expands can be modelled as a travelling wave, which evolve as solutions

to many reaction-diffusion equations. The models discussed in this report deal work under many

assumptions to simplify the situations, however this makes the wave speeds of their solutions much

easier to calculate.

All of the models analysed exhibit wave speeds of the form c = 2√εD with some variation. The

biggest deviation from this was the two species model wherein the invasive species would expand

their range front through the resident species territory at the speed c = 2

√ε1D1

(1− ε2µ12

ε1µ11

),

which simplifies to c = 2√εD when you set the resident species terms to zero. In the example

in Section 2.5 it is shown that even with a simplified model, the estimated expansion rate closely

matches the observed expansion rate, which indicates that these models are quite accurate despite

simplifications.

There are extensions to these models which I plan to investigate after this report; such as a

wider array of growth terms, heterogeneous environments with more than two patch types, systems

with three or more species present with different combinations of resident and invasive species, a

possible synergistic model whereby the invasive species actually spreads faster with the presence of

another species and potentially other mechanisms for diffusion.

27

References

[1] Shigesada, N., Kawasaki, K., 1997, Biological Invasions: Theory and Practice, Oxford Univer-

sity Press, Oxford

[2] Goel, N.S., Richter-Dyn, N., 1974, Stochastic Models in Biology, Academic Press, New York

[3] Fourier, J., 1822, Theorie analytique de la chaleur, Didot, Paris

[4] Skellam, J., 1951, ‘Random Dispersal in Theoretical Populations,’ Biometrika, Vol. 38, No.

1/2, pp. 196-218

[5] Ninti One Limited, 2013, Managing the impacts of feral camels across remote Australia:

Overview of the Australian Feral Camel Management Project, Report, viewed 19 February

2017, https://www.nintione.com.au/?p=3053.

[6] Fisher, R., 1937, ‘The Wave of Advance of Advantageous Genes,’ Annals of Eugenics, Vol. 7,

pp. 355-369

[7] Fife, P., 1979, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag Berlin

Heidelberg, New York

[8] Freeland, W.J., Martin, K.C., 1985, ‘The Rate of Range Expansion by Bufo marinus in North-

ern Australia, 1980-84’, Wildlife Research, Vol. 12, pp. 555-559.

[9] Lampo, M., Bayliss, P., 1996, ‘Density estimates of cane toads from native populations based

on mark-recapture data’, Wildlife Research, Vol. 23, pp. 305-315.

[10] Shigesada, N. et al., 1986, ‘Traveling Periodic Waves in Heterogeneous Environments,’ Theo-

retical Population Biology, Vol. 30, pp. 143-160

[11] Shigesada, N. et al., 1987, ‘The Speeds of Traveling Frontal Waves in Heterogeneous En-

vironments,’ Mathematical Topics in Population Biology, Morphogenesis and Neurosciences,

Springer-Verlag Berlin Heidelberg New York, pp. 88-97

28