mathematical theory of biological...
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VI Southern-Summer School on Mathematical Biology
Mathematical Theory of Biological Invasions
Part II
Sergei Petrovskii
Department of Mathematics, University of Leicester, UK
http://www.math.le.ac.uk/PEOPLE/sp237..
IFT/UNESP, Sao Paulo, January 16-27, 2017
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Plan of the course
• Introduction & a glance at field data
• Overview of mathematical tools
• Diffusion-reaction systemsI Single-species system: traveling waves, the problem of
critical domain, effects of environmental heterogeneityI Predator-prey system and the problem of biological control:
traveling waves and pattern formationI Beyond the traveling waves: patchy invasion
• Lattice models
• Kernel-based models (integro-difference equations):fat-tailed kernels, “superspread”, pattern formation
• Extensions, discussion, conclusions
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Plan of the course – Part II
• Introduction & a glance at field data
• Overview of mathematical tools
• Diffusion-reaction systems
Single-species system: traveling waves, the problem ofcritical domain, effects of environmental heterogeneity
Predator-prey system and the problem of biological control:traveling waves and pattern formation
Beyond the traveling waves: patchy invasion
• Lattice models
• Kernel-based models (integro-difference equations):fat-tailed kernels, “superspread”, pattern formation
• Extensions, discussion, conclusions
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Chapter V
Lattice models of biological invasion
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How essential is the choice of the model?
Specific questions:
• Is the patchy spread an artifact of the diffusion-reactionsystem?
• Concerns: Time-discrete framework may be moreappropriate, at least in some cases(e.g. for species with clearly different life stages)
• In order to take into account also the environmentheterogeneity, we now consider a system that is discreteboth in space and time
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How essential is the choice of the model?
Specific questions:
• Is the patchy spread an artifact of the diffusion-reactionsystem?
• Concerns: Time-discrete framework may be moreappropriate, at least in some cases(e.g. for species with clearly different life stages)
• In order to take into account also the environmentheterogeneity, we now consider a system that is discreteboth in space and time
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How essential is the choice of the model?
Specific questions:
• Is the patchy spread an artifact of the diffusion-reactionsystem?
• Concerns: Time-discrete framework may be moreappropriate, at least in some cases(e.g. for species with clearly different life stages)
• In order to take into account also the environmentheterogeneity, we now consider a system that is discreteboth in space and time
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Ecological example: metapopulation
(by Katrin Körner & Florian Jeltsch, University of Potsdam)
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In a more formal way:
(by Victoria Sork, UCLA)
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Coupled Map Lattice: single species
Continuous space (x , y) changes into a discrete ‘lattice’(xm, yn) where k = 1, . . . , M and n = 1, . . . , N.
Population numbers are defined only in the lattice nodes:
Each discrete step from t to t + 1 consists of distinctly differentdispersal stage and the ‘reaction’ stage.
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The dispersal stage includes emigration and immigration:
N′x ,y ,t = (1− µ)Nx ,y ,t +
∑(a,b)∈Vx,y
µ
4Na,b,t ,
where µ is the population fraction that emigrates from the site.
The choice of Vx ,y can be different, for instance
Vx ,y = (x − 1, y), (x + 1, y), (x , y − 1), (x , y + 1),
which corresponds to a certain ‘dispersal stencil’:
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The reaction stage is Nx ,y ,t+1 = f(
N′x ,y ,t
).
We assume that the population growth is hampered by thestrong Allee effect.
In particular, we consider
Nt+1 = f (Nt) =α (Nt)
2
1 + β2 (Nt)2 .
This function f (N) has two steady states, N∗1 and N∗
2 .
We also consider its approximation with a simpler function:
f (N) ≈ f (N) = N∗2H(N − N∗
1)
where H(z) is the Heaviside step function.
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Population growth in discrete time
N1* Prey Density0
N2*
Prey
Gro
wth
Rat
e
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Consider a single-site species introduction:
Questions to be answered:
• Under what conditions this introduction will lead tosuccessful establishment (and, possibly, spread)?
• What can be the rate of spread?
• What can be the pattern of spread?
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Establishment
The species will persist at the site p of initial introduction iff itssize after dispersal does not fall below the Allee threshold:
N′p = (1− µ)N∗
2 > N∗1 ,
that is, forµ < 1− κ where κ = N∗
1/N∗2 . (1)
The spread into a neighboring site q will be successful iff thedensity after dispersal exceeds the Allee threshold:
N′q =
µ
4N∗
2 > N∗1 ,
that is, forµ > 4κ. (2)
Conditions for establishment and spread are now not the same!
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Extinction-invasion diagram
κ* 1
µ
κ
1
0
I
II
III
IV
a b
Domain I - establishment & spread, Domain III - establishment without spread (invasion
pinning), Domain II - spread with pattern formation in the wake, Domain IV - extinction
(Mistro, Rodrigues & Petrovskii, 2012)
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Spread
For the step-like growth function, the rate of spread is exactly 1(one site per generation).
The shape of the envelope is an artefact of the dispersal stencil.
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Spread
But with a little bit of environmental heterogeneity...
Now the shape of the envelope looks much more realistic!
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Coupled Maps Lattice: predator-prey system
Now we have, for the dispersal stage
N′x ,y ,t = (1− µN)Nx ,y ,t +
∑(a,b)∈Vx,y
µN
4Na,b,t ,
P′x ,y ,t = (1− µP)Px ,y ,t +
∑(a,b)∈Vx,y
µP
4Pa,b,t ,
and for the reaction stage
Nx ,y ,t+1 = f(
N′x ,y ,t , P
′x ,y ,t
),
Px ,y ,t+1 = g(
N′x ,y ,t , P
′x ,y ,t
).
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Predator-prey on a lattice
Specifically, we choose the reaction term as follows
Nx ,y ,t+1 =r (Nx ,y ,t)
2
1 + b (Nx ,y ,t)2 · exp (−Px ,y ,t) ,
andPx ,y ,t+1 = Nx ,y ,tPx ,y ,t .
(in dimensionless variables) where N is prey and P is predator.
This system shows a very complicated dynamical behaviorincluding traveling waves, regular spatial patterns andspatiotemporal chaos.
(Mistro, Rodrigues & Petrovskii, 2012)
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Coupled Map Lattice: simulations
results in invasion failure. Once the rings are, eventually, pushed tothe domain boundary (Fig. 8e, f), both species go extinct. Since thedomain boundary is impenetrable (due to the no-flux condition)and the prey pulse is followed by the pulse of predators, there is noway to escape, so that finally all prey population is consumedwhich, in its turn, leads to the extinction of their specialistpredator. The corresponding total population size over time isshown in Fig. 12a.
A completely different regime of species spread is shown inFig. 9 (obtained for mN = 0.5 and mP = 0.8). In this case, nocontinuous traveling front is formed and the spread takes placethrough formation and movement of separate patches of highpopulation density. Correspondingly, we will call this regime the‘‘patchy invasion’’ (cf. Petrovskii et al., 2002a, 2005b; Morozovet al., 2006). The patch dynamics follows a complicated scenario:the patches can move, split, merge and split again, eventually
Fig. 7. Spatial distribution of prey for r = 4.5, b = 2, mN = 0.9, mP = 0.002 in time-step (a) t = 100, (b) t = 200 and (c) t = 2000. (d) Total population size of prey (solid curve) and
predator (dashed curve).
Fig. 8. Spatial distribution of prey at different moments: (a) t = 10, (b) t = 20, (c) t = 40, (d) t = 60, and (e) t = 80 and (f) t = 95 obtained for r = 4.2, b = 0.7, mN = 0.8 and mP = 0.5.
D.C. Mistro et al. / Ecological Complexity 9 (2012) 16–32 23
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Coupled Map Lattice: simulations
Prey Distribution Prey Distribution Prey Distribution
(a) (b) (c)Prey Distribution Prey Distribution Prey Distribution
(d) (e) (f)
Figure 11: Spatial distribution of prey in different time steps: (a) t = 25, (b) t = 35, (c) t = 50,
(d) t = 100, (e) t = 200 and (f) t = 235 for r = 4.2, b = 0.7, µN = 0.5 and µP = 0.8
only attractor) in order to explore the parameter range where patchy invasion can be
obtained.
We consider the following non-symmetric initial condition, which we will call Initial
Condition II: prey population is found inside the rectangle 48 ≤ x ≤ 53 and 47 ≤ y ≤
55 at its equilibrium value N∗
2 , while predator is initially present inside the rectangle
48 ≤ x ≤ 51 and 47 ≤ y ≤ 50 at P ∗. For three different combinations of the dynamical
parameters, namely r = 6. and b = 1.5, r = 4.2 and b = 0.7 and, r = 2.5 and b = 0.5, we
analyzed the invasion pattern for different dispersal parameter values.
The structure of the dispersal rate parameters regarding different scenarios of invasion
obtained are illustrated in Fig. 15. We observed extinction of both species, travelling
bands (Figure 16), patchy invasion (Fig. 17), transitional patterns of invasion (Fig. 18)
and wave fronts with chaos in the wake of invasion (Fig. 19) were obtained. Symbols
in Fig. 15 have the same meaning as before; circles stand for travelling bands. The
corresponding total prey and predator populations are illustrated in Figure (20).
Extinction occurs by the same mechanisms as for Initial Condition I and, for the sake
of brevity, we omit the illustrations of this case.
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Coupled Map Lattice: simulations
Prey Distribution Prey Distribution Prey Distribution
(a) (b) (c)Prey Distribution Prey Distribution Prey Distribution
(d) (e) (f)
Figure 21: Spatial distribution of prey in different time steps: (a) t = 50, (b) t = 100, (c)t = 150, (d) t = 200, (e) t = 250 and (f) t = 300 for r = 4.2, b = 0.7, µN = 0.3 and µP = 0.5.
as asymmetrical, coincident and non coincident prey and predator initial condition.
• Regarding biocontrol: Looking at the local dynamics, we would say that the predator
controls the prey (pest) in region C and can control in region F, depending on its
initial density. When space in added, we observed coexistence, in region C, for a
relatively broad region of the dispersal rates (see Fig. (19)). Predators lead the
prey population to extinction for dispersal rates indicated by filled diamonds in
Fig. (19)). On the other hand, when the species persist, the mean prey density is
dramatically smaller than it would be without predator.
In region A, the predator can control the prey invasion, depending on the area
where the predator is initially released. When both species are released only in the
central site, the range of the prey dispersal rate for which the invasion is successful
is smaller than that without the predator. (This simulation is not shown in the
text). However, if the prey range is initially larger than the predator, the control of
the prey is not observed.
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Coupled Map Lattice: simulations
Prey Distribution Prey Distribution Prey Distribution
(a) (b) (c)Prey Distribution Prey Distribution Prey Distribution
(d) (e) (f)
Figure 16: Spatial distribution of prey in different time steps: (a) t = 25, (b) t = 35, (c) t = 50,
(d) t = 100, (e) t = 150 and (f) t = 200 for r = 2.5, b = 0.5, µN = 0.6 and µP = 0.1.
Prey Distribution Prey Distribution Prey Distribution
(a) (b) (c)Prey Distribution Prey Distribution Prey Distribution
(d) (e) (f)
Figure 17: Spatial distribution of prey in different time steps: (a) t = 25, (b) t = 35, (c) t = 50,
(d) t = 100, (e) t = 150 and (f) t = 200 for r = 4.2, b = 0.7, µN = 0.2 and µP = 0.2.
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Chapter VI
Kernel-based (integral-difference)models of biological invasion
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Kernel-based models
Consider an insect population, e.g. moth, in a continuous spacebut with separated growth and dispersal stages:
Ut(x) → Ut = f (Ut(x)) → L(U) = Ut+1(x)
adult moth laid eggs, adult moth,
settling down larvae etc. new generation
where L is a spatial operator describing dispersal.
For simplicity, we consider dispersal at the infinite space.
Let k(x , y) is the probability distribution that a moth released atx will lay eggs at the position y , then
Ut+1(x) =
∫ ∞
−∞k(x , y)Ut(y)dy .
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Kernel-based models
Assume that space is homogeneous, k(x , y) → k(x − y).
We therefore obtain the following equation:
Ut+1(x) =
∫ ∞
−∞k(x − y)f (Ut(y))dy ,
where k(z) is also called the dispersal kernel.
Questions:
• How much different the kernel-based framework is fromdiffusion-reaction equations?
• If it is different, what can be the rate of spread?
The answer depends on the properties of the dispersal kernel.
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Examples of dispersal kernel
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Intuitively, the faster the rate of decay of k(z) at large z, thelower the rate of spread.
The properties of the kernel can be quantified by the behaviorof its moments.
The moment of the nth order:
mn =
∫ ∞
−∞znk(z)dz, m0 = 1, m1 =< z > .
For almost any k(z), mn is an increasing function of n.
However, a lot depends on how fast is the rate of increase.
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Case 1. All moments exist and the asymptotical rate ofincrease of mn is not faster than the factorial of n, i.e. at most
mn ∼ n!
which means that k(z) is exponentially bounded.
In this case, the kernel-based equation with compact initialconditions describes a traveling front propagating with aconstant speed (Lui 1983; Kot 1992)
The kernel-based model appears to be equivalent to thediffusion-reaction equation
(Petrovskii & Li, 2006, Section 2.2; Lewis et al., 2016, Section 2.4)
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Case 2. For a k(z) with a fatter tail (rate of decay lower thanexponential), the model has solutions of a new type:accelerating traveling waves.
The difference between the corresponding kernels can beexpressed in terms of the moment-generating function:
M(s) =
∫ ∞
−∞eszk(z)dz
(Kot et al. 1996), that is:
• Constant-speed traveling waves if M(s) exists
• Accelerating traveling waves if M(s) does not exist (theintegral diverges for any s 6= 0)
Accelerating waves do not exist if the population growth isdumped by the strong Allee effect
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Patterns in the wake
Interestingly, pattern formation in the wake of the traveling frontappears possible even in a single-species kernel-based model:
148 MARK ANDERSEN
A) cn
tI1; r_- -25 0 25
X
I D) 08
ZL <
0
1 O-50 l--!l -25 0 25
X
X X
FIG. 6. Results of first numerical experiment for the operator Q, of Equation (8).
(a) CY = 5, c = 1; (b) cx = 10, c = 1; Cc) (Y = 15, c = 1; (d) CY = 5, c = 2; (e) a = 10, c = 2; (f)
a = 15, c = 2.
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Questions arising
What can be the effect of other species?
How it may change the pattern of spread?
Consider a predator-prey system:
ut+1(r) =
∫Ω
k (u)(|r− r′|
)f(ut(r′), vt(r′))
dr′,
vt+1(r) =
∫Ω
k (v)(|r− r′|
)g(ut(r′), vt(r′))
dr′,
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Local demography: predator-prey system
ut+1(r) =r (ut(r))2
1 + b (ut(r))2 · exp (−vt(r)) ,
vt+1(r) = ut(r)vt(r) .
1
3
6
5
2
4
0 1 2 3 4 5 6 7
0
2
4
6
8
10
b
a
(Mistro, Rodrigues & Petrovskii, 2012)
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Local demography: predator-prey system
ut+1(r) =r (ut(r))2
1 + b (ut(r))2 · exp (−vt(r)) ,
vt+1(r) = ut(r)vt(r) .
1
3
6
5
2
4
0 1 2 3 4 5 6 7
0
2
4
6
8
10
b
a
(Mistro, Rodrigues & Petrovskii, 2012)
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Local demography: predator-prey system
ut+1(r) =r (ut(r))2
1 + b (ut(r))2 · exp (−vt(r)) ,
vt+1(r) = ut(r)vt(r) .
1
3
6
5
2
4
0 1 2 3 4 5 6 7
0
2
4
6
8
10
b
a
(Mistro, Rodrigues & Petrovskii, 2012)
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Dispersal kernel: the “reference case”
kG
(|r− r′|
)=
12πα2
iexp
(−|r− r′|2
2α2i
).
Dispersal with the Gaussian kernel is known to be equivalent(in some sense) to diffusion.
10 0 10
10
0
10
PreyDistribut
10 0 10
10
0
10
PreyDistribut
t=140 t=200
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Fat-tailed kernels in 1D
Long-distance asymptotics for the Gaussian kernel:
k(x) ∼ e−ax2.
Fat tailed kernel – power-law decay:
k(x) ∼ x−µ (1 < µ < 3)
In case µ = 2, the stable distribution is available in a closedform known as Cauchy distribution:
kC(x) =β
π(β2 + x2)∼ x−2.
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Fat-tailed kernels in 2D
Long-distance asymptotics: k(r) ∼ r−(µ+1) (1 < µ < 3)
Explicit form of the stable distribution is not available, henceextension onto the 2D case is ambiguous.
Cauchy kernels Type I:
kCI (r, r′) =β2
iπ(βi + |r− r′|)3 ∼ |r− r′|−3 ,
Cauchy kernels Type II:
kCII (r, r′) =γi
2π(γ2
i + |r− r′|2)3/2 ∼ |r− r′|−3 .
(Rodrigues et al., 2015)
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Fat-tailed kernels
Cauchy kernel has significantly different properties compared tothe Gaussian kernel: the variance does not exist, < r2 >= ∞.
• The fact that < r2 >= ∞ is sometimes interpreted as theinfinite correlation length
• Invasive species can spread with an accelerating speed(Kot et al. 1996)
Questions arising:
• Can patchy spread occur for the fat-tailed dispersal?
• How the rate of spread may differ between differentkernels?
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Simulations, kernel Type I
-30 0 30
-30
0
30
Prey Distribution
-20
-20
Prey Distribution
t = 20 t = 100
-30 0 30
-30
0
30
Prey Distribution
-30 0 30
-30
0
30
Prey Distribution
t = 140 t = 190
Figure 1: Snapshots of the prey spatial distribution at different moments t, (a) 20, (b) 100, (c)140, and (d) 190, as obtained for the Cauchy kernel Type I and the asymmetrical initial conditions(??–??). Parameters are βN = 0.0488 and βP = 0.098.
2
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Simulations, kernel Type II
-30 0 30
-30
0
30
Prey Distribution
-30 0 30
-30
0
30
Prey Distribution
t = 40 t = 80
-30 0 30
-30
0
30
Prey Distribution
-30 0 30
-30
0
30
Prey Distribution
t = 140 t = 200
Figure 2: Snapshots of the prey spatial distribution at different moments t, (a) 40, (b) 80, (c) 140and (d) 200, as obtained for the Cauchy kernel Type II and the asymmetrical initial conditions(??–??). Parameters are γN = 0.068 and γP = 0.1205.
3
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How can we compare the results for different dispersal kernels,i.e. Gaussian, Cauchy Type I and Cauchy Type II ?
Standard approach (equating the variances) does not work asthe variance does not exist – “scale-free” process
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Conditions of equivalence
Consider radius ε within which the probability of finding anindividual after dispersal is 1/2:
Pε =
∫ ∫|r|≤ε
dr =
∫ 2π
0
∫ ε
0ki(r , θ)rdrdθ =
12.
For the Gaussian kernel, we obtain ε = α√
2 ln 2.
For Cauchy kernel Type I:
β = ε(√
2− 1) = α(2−√
2)√
ln 2 ≈ 0.4877α.
For Cauchy kernel Type II:
γ =ε√3
= α
√23
ln 2 ≈ 0.6798α.
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Radius of invaded area vs time
0 50 100 150 2000
5
10
15
20
25
Time
PreyWavefron
0 50 100 150 2000
5
10
15
20
Time
PreyWavefron
Cauchy Type I Cauchy Type II
Invasion rates are related by the above equivalence condition.
There is no accelerated spread.
Invasion rates obtained for the Cauchy kernels are between1-10 km/year, hence in excellent agreement with field data.
(Rodrigues et al., 2015)
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• This is the end of the course...
• But certainly not the end of the story
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• This is the end of the course...
• But certainly not the end of the story
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Literature cited in the notesLewis, M.A., Petrovskii, S.V. & Potts, J. (2016) The Mathematics Behind BiologicalInvasions. Interdisciplinary Applied Mathematics, Vol. 44. Springer.Petrovskii, S.V. & Li, B.-L. (2006) Exactly Solvable Models of Biological Invasion,Chapman & Hall / CRC Press. (pdf is available on my website.)Owen, M.R. & M.A. Lewis (2001). How predation can slow, stop or reverse a preyinvasion. Bull. Math. Biol. 63, 655-684.Petrovskii, S.V., Malchow, H. & Li B.-L. (2005) An exact solution of a diffusivepredator-prey system. Proc. R. Soc.Lond. A 461, 1029-1053.Sherratt, J.A., Lewis, M.A. & Fowler, A.C. (1995) Ecological chaos in the wake ofinvasion. Proc. Natl. Acad. Sci. USA 92, 2524-2528.Jankovic, M. & Petrovskii, S. (2013) Gypsy moth invasion in North America: Asimulation study of the spatial pattern and the rate of spread. Ecol. Compl. 14,132-144.Mistro, D.C., Rodrigues, L.A.D. & Petrovskii, S.V. (2012) Spatiotemporal complexity ofbiological invasion in a space- and time-discrete predator-prey system with the strongAllee effect. Ecological Complexity 9, 16-32.Rodrigues, L.A.D., Mistro, D.C., Cara, E.R., Petrovskaya, N. & Petrovskii, S.V. (2015)Patchy invasion of stage-structured alien species with short-distance and long-distancedispersal. Bull. Math. Biol. 77, 1583-1619.Kot, M., Lewis, M. A. & van der Driessche, P. (1996) Dispersal data and the spread ofinvading organisms. Ecology 77, 2027-2042.
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Other useful references
Shigesada, N. & Kawasaki, K. (1997) Biological Invasions: Theory and Practice.Oxford University Press.
Lewis, M.A. & Kareiva, P. (1993). Allee dynamics and the spread of invadingorganisms. Theor. Popul. Biol. 43, 141-158.
Neubert, M.G., Kot, M. & Lewis, M.A. (1995) Dispersal and pattern formation in adiscrete-time predator-prey model. Theor. Pop. Biol. 48, 7-43.
Lui, R. (1983) Existence and stability of travelling wave solutions of a nonlinear integraloperator. J. Math. Biol. 16, 199-220.
Volpert, V. & Petrovskii, S.V. (2009) Reaction-diffusion waves in biology. Physics of LifeReviews 6, 267-310.
Kot, M. (2001) Elements of Mathematical Ecology. Cambridge University Press.
Malchow, H., Petrovskii, S.V. & Venturino, E. (2008) Spatiotemporal Patterns in Ecologyand Epidemiology: Theory, Models, Simulations. Chapman & Hall / CRC Press.
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Good luck with your research!