IFT/UNESP,SaoPaulo,January16-27,2017 SergeiPetrovskii PartII MathematicalTheoryofBiologicalInvasions

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

Plan of the course

• Introduction & a glance at field data

• Overview of mathematical tools

• Diffusion-reactionsystems

I Single-species system: traveling waves, the problem of critical domain, effects of environmental heterogeneity

I Predator-prey system and the problem of biological control:

traveling waves and pattern formation

I Beyond the traveling waves: patchy invasion

• Lattice models

• Kernel-based models(integro-difference equations):

fat-tailed kernels, “superspread”, pattern formation

• Extensions, discussion, conclusions

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 of critical 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

Chapter V

Lattice models of biological invasion

How essential is the choice of the model?

Specific questions:

• Is the patchy spread an artifact of the diffusion-reaction system?

• Concerns: Time-discrete framework may be more appropriate, at least in some cases

(e.g. for species with clearly differentlife stages)

• In order to take into account also the environment heterogeneity, we now consider a system that isdiscrete both in space and time

How essential is the choice of the model?

Specific questions:

• Is the patchy spread an artifact of the diffusion-reaction system?

• Concerns: Time-discrete framework may be more appropriate, at least in some cases

(e.g. for species with clearly differentlife stages)

• In order to take into account also the environment heterogeneity, we now consider a system that isdiscrete both in space and time

How essential is the choice of the model?

Specific questions:

• Is the patchy spread an artifact of the diffusion-reaction system?

• Concerns: Time-discrete framework may be more appropriate, at least in some cases

(e.g. for species with clearly differentlife stages)

• In order to take into account also the environment heterogeneity, we now consider a system that isdiscrete both in space and time

Ecological example: metapopulation

(by Katrin Körner & Florian Jeltsch, University of Potsdam)

In a more formal way:

(by Victoria Sork, UCLA)

A possible mathematical framework: discrete space, continuous time (Keitt et al., 2001)

Coupled Map Lattice: single species

Continuous space(x,y)changes into a discrete ‘lattice’

(x_m,y_n)wherek =1, . . . ,M andn=1, . . . ,N.

Population numbers are defined only in the lattice nodes:

Each discrete step fromt tot+1 consists of distinctly different dispersal stageand the‘reaction’ stage.

Thedispersal stageincludes emigration and immigration:

N_x,y,t⁰ = (1−µ)Nx,y,t + X

(a,b)∈Vx,y

µ 4N_a,b,t ,

whereµis the population fraction that emigrates from the site.

The choice ofVx,y can be different, for instance

V_x,y ={(x−1,y),(x+1,y),(x,y−1),(x,y +1)}, which corresponds to a certain ‘dispersal stencil’:

Thereaction stageis N_x,y,t+1=f N_x,y,t⁰

.

We assume that the population growth is hampered by the strong Allee effect.

In particular, we consider

N_t+1=f(N_t) = α(N_t)² 1+β²(N_t)². This functionf(N)has two steady states,N₁^∗ andN₂^∗. We also consider itsapproximationwith a simpler function:

f(N)≈˜f(N) =N₂^∗H(N−N₁^∗) whereH(z)is the Heavisidestep function.

Population growth in discrete time

N₁^* Prey Density 0

N₂^*

Prey Growth Rate

Consider a single-site species introduction:

Questions to be answered:

• Under whatconditionsthis introduction will lead to successful establishment(and, possibly, spread)?

• What can be therate of spread?

• What can be thepattern of spread?

Establishment

The species willpersistat the sitepof initial introduction iff its size after dispersal does not fall below the Allee threshold:

N_p⁰ = (1−µ)N₂^∗ >N₁^∗, that is,for

µ <1−κ where κ=N₁^∗/N₂^∗ . (1) Thespreadinto a neighboring siteqwill be successful iff the density after dispersal exceeds the Allee threshold:

N_q⁰ = µ

4N₂^∗ >N₁^∗, that is,for

µ >4κ. (2)

Conditions for establishment and spread are nownot the same!

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 withpattern formationin the wake, Domain IV -extinction

(Mistro, Rodrigues & Petrovskii, 2012)

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.

Spread

But with a little bit of environmental heterogeneity...

Now the shape of the envelope looks much more realistic!

Coupled Maps Lattice: predator-prey system

Now we have, for thedispersal stage N_x,y,t⁰ = (1−µ_N)N_x,y,t + X

(a,b)∈Vx,y

µ_N

4 N_a,b,t , P_x,y,t⁰ = (1−µP)P_x,y,t + X

(a,b)∈Vx,y

µP

4 P_a,b,t ,

and for thereaction stage N_x,y,t+1=f

N_x,y,t⁰ ,P_x,y,t⁰ ,

P_x,y,t+1=g

N_x,y,t⁰ ,P_x,y⁰ _,t .

Predator-prey on a lattice

Specifically,we choosethe reaction term as follows N_x,y,t+1= r(N_x,y,t)²

1+b(Nx,y,t)² ·exp(−P_x,y,t), and

P_x,y,t+1=Nx,y,tPx,y,t .

(in dimensionless variables) whereNis prey andP is predator.

This system shows a verycomplicated dynamical behavior including traveling waves, regular spatial patterns and spatiotemporal chaos.

(Mistro, Rodrigues & Petrovskii, 2012)

Coupled Map Lattice: simulations

resultsininvasionfailure.Oncetheringsare,eventually,pushedto thedomainboundary(Fig.8e,f),bothspeciesgoextinct.Sincethe domainboundaryisimpenetrable(duetotheno-fluxcondition) andthepreypulseisfollowedbythepulseofpredators,thereisno waytoescape,sothatfinallyallpreypopulationisconsumed which,in itsturn,leadsto theextinctionoftheir specialist predator.Thecorrespondingtotalpopulationsizeovertimeis showninFig.12a.

Acompletelydifferentregimeofspeciesspreadisshownin Fig.9 (obtained for mN=0.5 and mP=0.8). In thiscase, no continuoustravelingfrontisformedandthespreadtakesplace throughformationandmovementofseparatepatchesofhigh populationdensity.Correspondingly,wewillcallthisregimethe

‘‘patchyinvasion’’(cf.Petrovskiietal.,2002a,2005b;Morozov etal.,2006).Thepatchdynamicsfollowsacomplicatedscenario:

thepatchescanmove,split,mergeandsplitagain,eventually Fig.7.Spatialdistributionofpreyforr=4.5,b=2,mN=0.9,mP=0.002intime-step(a)t=100,(b)t=200and(c)t=2000.(d)Totalpopulationsizeofprey(solidcurve)and predator(dashedcurve).

Fig.8.Spatialdistributionofpreyatdifferentmoments:(a)t=10,(b)t=20,(c)t=40,(d)t=60,and(e)t=80and(f)t=95obtainedforr=4.2,b=0.7,mN=0.8andmP=0.5.

D.C.Mistroetal./EcologicalComplexity9(2012)16–32 23

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 forr= 4.2,b= 0.7,µN = 0.5 andµP= 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 N₂^∗, while predator is initially present inside the rectangle 48≤x≤51 and 47≤y≤50 atP^∗. For three different combinations of the dynamical parameters, namelyr= 6.andb= 1.5,r= 4.2 andb= 0.7 and,r= 2.5 andb= 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.

11

This patchy invasion occurs in the parameter range where the nonspatial system goes extinct

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 forr= 4.2,b= 0.7,µN= 0.3 andµP= 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.

21

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 forr= 2.5,b= 0.5,µN = 0.6 andµP= 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 forr= 4.2,b= 0.7,µN = 0.2 andµP= 0.2.

14

Chapter VI

Kernel-based (integral-difference)

models of biological invasion

Kernel-based models

Consider an insect population, e.g. moth, in acontinuousspace but withseparated growthanddispersalstages:

U_t(x) → U˜_t =f(U_t(x)) → L( ˜U) =U_t+1(x)

adult moth laid eggs, adult moth,

settling down larvae etc. new generation

whereLis a spatial operator describing dispersal.

For simplicity, we consider dispersal at the infinite space.

Letk(x,y)is theprobability distributionthat a moth released at x will lay eggs at the positiony, then

U_t+1(x) = Z ∞

−∞

k(x,y) ˜Ut(y)dy .

Kernel-based models

Assume that space ishomogeneous,k(x,y)→k(x −y).

We therefore obtain the following equation:

U_t+1(x) = Z ∞

−∞

k(x −y)f(Ut(y))dy ,

wherek(z)is also called thedispersal kernel.

Questions:

• How much differentthe kernel-based framework is from diffusion-reaction equations?

• If it is different, what can be therate of spread?

The answer depends on the properties of the dispersal kernel.

Examples of dispersal kernel

Intuitively, thefasterthe rate of decay ofk(z)at largez, the lowerthe rate of spread.

The properties of the kernel can be quantified by the behavior of itsmoments.

The moment of thenth order:

m_n= Z ∞

−∞

zⁿk(z)dz, m₀=1, m₁=<z > .

For almost anyk(z),m_nis anincreasing function ofn.

However, a lot depends on how fast is the rate of increase.

(e.g. see Kot et al., 1996)

Case 1. All moments exist and the asymptotical rate of increase ofm_nis not faster than the factorial ofn, i.e. at most

m_n∼n!

which means thatk(z)isexponentially bounded.

In this case, the kernel-based equation with compact initial conditions describes atraveling frontpropagating with a constant speed(Lui 1983; Kot 1992)

The kernel-based model appears to beequivalentto the diffusion-reaction equation

(Petrovskii & Li, 2006, Section 2.2; Lewis et al., 2016, Section 2.4)

Case 2. For ak(z)with afatter tail(rate of decay lower than exponential), the model has solutions of a new type:

acceleratingtraveling waves.

The difference between the corresponding kernels can be expressed in terms of themoment-generating function:

M(s) = Z ∞

−∞

e^szk(z)dz

(Kot et al. 1996), that is:

• Constant-speed traveling waves ifM(s)exists

• Accelerating traveling waves ifM(s)does not exist (the integral diverges for anys6=0)

Accelerating wavesdo not exist ifthe population growth is dumped by thestrong Allee effect

Patterns in the wake

Interestingly, pattern formation in the wake of the traveling front appears possible even in asingle-specieskernel-based model:

148 MARK ANDERSEN

A)

cn

tI1; ^r_-

-25 0 25

X

I

08

ZL <

0

1 O-50

l--!l

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.

(Andersen, 1991)

Questions arising

What can be theeffect of other species?

How it may change the pattern of spread?

Consider apredator-preysystem:

u_t+1(r) = Z

Ω

k^(u) |r−r⁰|

f ut r⁰

,vt r⁰ dr⁰,

v_t+1(r) = Z

Ω

k^(v) |r−r⁰|

g u_t r⁰

,v_t r⁰ dr⁰,

Local demography: predator-prey system

u_t+1(r) = r(u_t(r))²

1+b(u_t(r))² ·exp(−v_t(r)), v_t+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)

Local demography: predator-prey system

u_t+1(r) = r(u_t(r))²

1+b(u_t(r))² ·exp(−v_t(r)), v_t+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)

Limit cycle

Local demography: predator-prey system

u_t+1(r) = r(u_t(r))²

1+b(u_t(r))² ·exp(−v_t(r)), v_t+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)

Extinction in the nonspatial system

Dispersal kernel: the “reference case”

k_G

|r−r^0|

= 1

2πα²_i exp −|r−r⁰|² 2α²_i

! .

Dispersal with the Gaussian kernel is known to beequivalent (in some sense)to diffusion.

10 0 10

10 0 10

PreyDistribut

10 0 10

10 0 10

PreyDistribut

t=140 t=200

Fat-tailed kernels in 1D

Long-distance asymptotics for the Gaussian kernel:

k(x)∼e^−ax2.

Fat tailed kernel –power-lawdecay:

k(x)∼x^−µ (1< µ <3)

In caseµ=2, thestable distributionis available in a closed form known asCauchy distribution:

k_C(x) = β

π(β²+x²) ∼ x⁻².

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, hence extension onto the 2D case is ambiguous.

Cauchy kernelsType I:

k_CI(r,r⁰) = β_i²

π(βi+|r−r⁰|)³ ∼ |r−r⁰|⁻³,

Cauchy kernelsType II:

k_CII(r,r⁰) = γ_i

2π γ_i²+|r−r⁰|²3/2 ∼ |r−r⁰|⁻³.

(Rodrigues et al., 2015)

Fat-tailed kernels

Cauchy kernel has significantly different properties compared to the Gaussian kernel: thevariance does not exist,<r²>=∞.

• The fact that<r²>=∞is sometimes interpreted as the infinite correlation length

• Invasive species can spread with an accelerating speed (Kot et al. 1996)

Questions arising:

• Can patchy spreadoccurfor the fat-tailed dispersal?

• How therate of spreadmay differ between different kernels?

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 momentst, (a) 20, (b) 100, (c) 140, and (d) 190, as obtained for the Cauchy kernel Type I and the asymmetrical initial conditions (??–??). Parameters areβ_N= 0.0488 andβ_P= 0.098.

2

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 momentst, (a) 40, (b) 80, (c) 140 and (d) 200, as obtained for the Cauchy kernel Type II and the asymmetrical initial conditions (??–??). Parameters areγ_N= 0.068 andγ_P= 0.1205.

3

How can wecompare the resultsfor different dispersal kernels, i.e. Gaussian, Cauchy Type I and Cauchy Type II ?

Standard approach (equating the variances) does not work as the variance does not exist – “scale-free” process

Conditions of equivalence

Consider radiuswithin which theprobability of finding an individualafter dispersal is 1/2:

P = Z Z

|r|≤

dr= Z 2π

0

Z

0

k_i(r, θ)rdrdθ= 1 2. For the Gaussian kernel, we obtain =α√

2 ln 2.

For Cauchy kernel Type I:

β=(

√

2−1) =α(2−√ 2)

√

ln 2≈0.4877α.

For Cauchy kernel Type II:

γ=

√ 3 =α

r2

3ln 2≈0.6798α.

Radius of invaded area vs time

0 50 100 150 200

0 5 10 15 20 25

Time

PreyWavefron

0 50 100 150 200

0 5 10 15 20

Time

PreyWavefron

Cauchy Type I Cauchy Type II

Invasion rates are related by the above equivalence condition.

There isno accelerated spread.

Invasion rates obtained for the Cauchy kernels are between 1-10 km/year, hence inexcellent agreementwith field data.

(Rodrigues et al., 2015)

•

This is the end of the course...

•

But certainly not the end of the story

•

This is the end of the course...

•

But certainly not the end of the story

Literature cited in the notes

Lewis, M.A., Petrovskii, S.V. & Potts, J. (2016)The Mathematics Behind Biological Invasions. 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 prey invasion.Bull. Math. Biol.63, 655-684.

Petrovskii, S.V., Malchow, H. & Li B.-L. (2005) An exact solution of a diffusive predator-prey system.Proc. R. Soc.Lond. A461, 1029-1053.

Sherratt, J.A., Lewis, M.A. & Fowler, A.C. (1995) Ecological chaos in the wake of invasion.Proc. Natl. Acad. Sci. USA92, 2524-2528.

Jankovic, M. & Petrovskii, S. (2013) Gypsy moth invasion in North America: A simulation 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 of biological invasion in a space- and time-discrete predator-prey system with the strong Allee effect.Ecological Complexity9, 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-distance dispersal.Bull. Math. Biol.77, 1583-1619.

Kot, M., Lewis, M. A. & van der Driessche, P. (1996) Dispersal data and the spread of invading organisms.Ecology77, 2027-2042.

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 invading organisms.Theor. Popul. Biol.43, 141-158.

Neubert, M.G., Kot, M. & Lewis, M.A. (1995) Dispersal and pattern formation in a discrete-time predator-prey model.Theor. Pop. Biol.48, 7-43.

Lui, R. (1983) Existence and stability of travelling wave solutions of a nonlinear integral operator.J. Math. Biol.16, 199-220.

Volpert, V. & Petrovskii, S.V. (2009) Reaction-diffusion waves in biology.Physics of Life Reviews6, 267-310.

Kot, M. (2001)Elements of Mathematical Ecology.Cambridge University Press.

Malchow, H., Petrovskii, S.V. & Venturino, E. (2008)Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations.Chapman & Hall / CRC Press.

Keitt, T.H., Lewis, M.A., Holt R.D. (2001) Allee effects, invasion pinnings, and species' borders. Am. Nat. 157, 203-216.

Good luck with your research!