Biological Invasion: Observations, Theory, Models, Simulations

Transcription

1 Biological Invasion: Observations, Theory, Models, Simulations Sergei Petrovskii Department of Mathematics University of Leicester, UK. Midlands MESS: Leicester, March 13, 212

2 Plan of the Talk Introduction A glance at field data Mathematical model I: reaction-diffusion Mathematical model II: coupled maps Case study: gypsy moth spread in North America Conclusions & open question

3 Introduction: What it is all about The term biological invasion is a common name for a variety of phenomena related to introduction and spatial spread of alien or exotic species, i.e., species that have not been present in a given ecosystem until they are brought in. Consequences of species invasion: 1. A new species often becomes a pest and that can result in huge economic losses. For instance, economic loss from the invasion of insect pests in the USA for the period from 196 to 1991 is estimated to be $ 92 billion (U.S. Office of Technology Assessment, 1993). 2. Severe damage to biodiversity.

4 (c) Spread of the introduced species over space, invasion Stages of biological invasion Stages of biological invasion. (a) Introduction of an alien species: (b) Establishment of the introduced species in the new environment: U? x

5 Stages of biological invasion (contd.) x (c) Spread of the introduced species over space, invasion of new areas: U where U is the population density. x

6 Stages of biological invasion (contd.) x (c) Spread of the introduced species over space, invasion of new areas: U where U is the population density. x Is the pattern always as simple as the travelling front?

7 Stages of biological invasion (contd.) x (c) Spread of the introduced species over space, invasion of new areas: U where U is the population density. x Is the pattern always as simple as the travelling front? If not, why not, and what can come instead?

8 Example of historical data Ecological examples and applications 173 FIGURE 8.1: Geographical spread of Japanese beetle in the United States from the place of its original introduction. Alternating grey and black colors show the areas invaded during successive time intervals (from United States Bureau of Entomology and Plant Quarantine, 1941). Invasion of Japanese beetle in the United States (approx to 1945) An attempt to explain the nature of accelerating waves has been made by means of linking them to non-gaussian diffusion when every single event of dispersal follows not the normal distribution but a distribution with much slower rate of decay at large distances (see Section 2.2). Mathematically, it means that the models should be based on integral-difference equations, not

9 Field data: spread of gypsy moth in the USA The red spots show heavily infested areas (from the US Forest Service,

10 Factors that may affect the pattern of spread Heterogeneity of environment

11 Factors that may affect the pattern of spread Heterogeneity of environment Human-assisted dispersal (aka stratified diffusion )

12 Factors that may affect the pattern of spread Heterogeneity of environment Human-assisted dispersal (aka stratified diffusion ) Predation?

13 Factors that may affect the pattern of spread Heterogeneity of environment Human-assisted dispersal (aka stratified diffusion ) Predation? Impact of pathogens?

14 Factors that may affect the pattern of spread Heterogeneity of environment Human-assisted dispersal (aka stratified diffusion ) Predation? Impact of pathogens?...?

15 Factors that may affect the pattern of spread Heterogeneity of environment Human-assisted dispersal (aka stratified diffusion ) Predation? Impact of pathogens?...?

16 Mathematical model: A predator-prey system U(R, T ) = D 1 2 U(R, T ) + f (U)U r(u)v, T V (R, T ) = D 2 2 V (R, T ) + κr(u)v MV T where U, V are the densities of prey and predator, respectively, at position R = (X, Y ) and time T.

17 Mathematical model: A predator-prey system U(R, T ) = D 1 2 U(R, T ) + f (U)U r(u)v, T V (R, T ) = D 2 2 V (R, T ) + κr(u)v MV T where U, V are the densities of prey and predator, respectively, at position R = (X, Y ) and time T. Additionally, we include: Holling type II for predation: r(u) = η U H+U A strong Allee effect for prey: f (U) = α(u U )(K U)

18 Mathematical model (contd.) In dimensionless variables: u(r, t) t v(r, t) t = 2 u(r, t) + γu(u β)(1 u) uv 1 + Λu, = ɛ 2 v(r, t) + uv 1 + Λu mv. Note that, for Λ =, these equations describe also a special case of SI model of a host-pathogen system.

19 1D case Theory. Without the Allee effect, the generic pattern of species spread is predicted to be a travelling front. However, travelling waves in a predator-prey system with the Allee effect are poorly studied. Simulations. Consider the following initial conditions: u(x, ) = u for u < x < u, otherwise u(x, ) =, v(x, ) = v for v < x < v, otherwise v(x, ) =, where u, v are the initial population densities, u and v give the radius of the initially invaded domain.

20 1D case (contd.) 42 S.V. Petrovskii, H. Malchow / Nonlinear Analysis: Real World Applications 1 (2) Fig. 2. The density of the predators vs. coordinate calculated for equidistant time moments with t = 16 for k = 2:; p = :4; h = :6. The rectangle at the lower left corner shows the initial distribution, the arrow indicates the direction of the population wave propagation.

21 1D case (contd.) 1.2 c Population Density c o Space

22 1D case (contd.) Population Density Space

23 1D case (contd.) Population Density Space

24 1D case (contd.) Population Density Space

25 Regimes of biological invasion in a predator-prey system Regimes of extinction 1. Ordinary extinction Regimes of local invasion 2. Dynamical localization 3. Patchy extinction Regimes of geographical spread Travelling population pulses 1. Stationary pulses 2. Oscillating pulses Travelling population fronts Patchy spread Smooth population fronts Population fronts with spatiotemporal patterns in the wake

26 2D case, simulations The initial conditions: u(x, y, ) = u for x 11 < x < x 12, y 11 < y < y 12, u(x, y, ) = otherwise, v(x, y, ) = v for x 21 < x < x 22, y 21 < y < y 22, v(x, y, ) = otherwise.

27 3 a) t= b) 1 2 t= c) t= Figure 1

28 2D case (contd.) 3 (a) 3 (b) Space, y 2 1 Space, y Space, x Space, x 3 (c) 3 (d) Space, y 2 1 Space, y Space, x Space, x

29 2D case (contd.) 3 (a) 3 (b) Space, y 2 1 Space, y Space, x Space, x 3 (c) 3 (d) Space, y 2 1 Space, y Space, x Space, x

30 2D case (contd.) 3 (a) 3 (b) Space, y 2 1 Space, y Space, x Space, x 3 (c) 3 (d) Space, y 2 1 Space, y Space, x Space, x

31 2D case (contd.) 3 (a) 3 (b) Space, y 2 1 Space, y Space, x Space, x 3 (c) 3 (d) Space, y 2 1 Space, y Space, x Space, x

32 Invasion at the edge of extinction Patchy invasion Invasion through propagation of continuous travelling fronts Extinction Predation

33 Without space: Predation strength increases from left to right: Predator Density Predator Density Prey Density Prey Density Patchy spread occur when the nonspatial system is not viable!

34 ...And now something completely different How essential is the choice of the model? Is the patchy spread an artifact of the diffusion-reaction system?

35 ...And now something completely different How essential is the choice of the model? Is the patchy spread an artifact of the diffusion-reaction system? Concerns: Time-discrete framework may be more appropriate, at least in some cases

36 ...And now something completely different How essential is the choice of the model? Is the patchy spread an artifact of the diffusion-reaction system? Concerns: Time-discrete framework may be more appropriate, at least in some cases In order to take into account also the environment heterogeneity, we now consider a system that is discrete both in space and time

37 Coupled Map Lattice Continuous space (x, y) changes into a discrete lattice (x m, y n ) where k = 1,..., M and n = 1,..., N. Each discrete step from t to t + 1 consists of distinctly different dispersal stage and the reaction stage. The dispersal stage is described as N x,y,t = (1 µ N )N x,y,t + µ N 4 N a,b,t, (a,b) V x,y P x,y,t = (1 µ P )P x,y,t + µ P 4 P a,b,t, (a,b) V x,y where V x,y = {(x 1, y), (x + 1, y), (x, y 1), (x, y + 1)}.

38 Coupled Map Lattice (contd.) In it turn, the reaction stage is described as ( ) N x,y,t+1 = f N x,y,t, P x,y,t, ( ) P x,y,t+1 = g N x,y,t, P x,y,t. Specifically, we choose (in dimensionless variables) and N x,y,t+1 = r (N x,y,t) b (N x,y,t ) 2 exp ( P x,y,t), P x,y,t+1 = N x,y,t P x,y,t.

39 Coupled Map Lattice: simulations Prey Distribution Prey Distribution Prey Distribution (a) (b) (c) Prey Distribution Prey Distribution Prey Distribution (d) (e) (f)

40 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 = 5, (b) t = 1, (c)

41 Figure 17: Spatial distribution of prey in different time steps: (a) t = 25, (b) t = 35, (c) t = 5, Coupled Figure 16: Spatial Map distribution Lattice: of preysimulations in different time steps: (a) t = 25, (b) t = 35, (c) t = 5, (d) t = 1, (e) t = 15 and (f) t = 2 for r = 2.5, b =.5, µ N =.6 and µ P =.1. Prey Distribution Prey Distribution Prey Distribution (a) (b) (c) Prey Distribution Prey Distribution Prey Distribution (d) (e) (f)

42 Gypsy moth spread revisited: Biological evidence Environmental heterogeneity is not sufficient to explain the patchy spread (Sharov et al. 1997; Whitmire & Tobin 26) Currently, a plausible explanation is seen in the human-assisted dispersal (Liebhold et al. 1992; Liebhold & Tobin 26) However, here we argue that its contribution can be rather exaggerated: Long-distance dispersal is a rare event New colonies will normally be wiped out by the strong Allee effect

43 Gypsy moth spread revisited: Biological evidence Existence of the strong Allee effect (Liebhold & Bascompte 23; Whitemire & Tobin 26; Liebhold & Tobin 21) Predation: not significant (Elkinton et al. 1996, 24) Infection by multicapsid nuclear polyhedrosis virus (NPV): significant (Elkinton & Liebhold 199; Reardon et al. 29)

44 Gypsy moth spread revisited: Biological evidence Existence of the strong Allee effect (Liebhold & Bascompte 23; Whitemire & Tobin 26; Liebhold & Tobin 21) Predation: not significant (Elkinton et al. 1996, 24) Infection by multicapsid nuclear polyhedrosis virus (NPV): significant (Elkinton & Liebhold 199; Reardon et al. 29) SI model of pathogen spread: s(r, t) t i(r, t) t = ɛ 2 s(r, t) + γs(s β)(1 s) si, = 2 i(r, t) + si mi, s is the density of susceptible insects and i is the density of infected

45 Summary of simulation results Mortality of infected population, m Extinction Continuous fronts patchy invasion continuous fronts extinction transitional Growth rate of susceptible population, γ

46 Distorted continuous front

47 Transitional regime

48 Patchy invasion

49 Theory against data The spatial scales and the rate of spread obtained in simulations are consistent with those observed in field data

50 Conclusions Interplay between the Allee effect and predation or infection can turn continuous-front invasion into a patchy invasion. This is a generic property and can be observed in continuous systems as well as in discrete ones. Theoretical results are consistent with (some) field data

51 Conclusions Interplay between the Allee effect and predation or infection can turn continuous-front invasion into a patchy invasion. This is a generic property and can be observed in continuous systems as well as in discrete ones. Theoretical results are consistent with (some) field data These results seem to lead to a new paradigm of invasion pattern: patchy spread instead of continuous front

52 Conclusions Interplay between the Allee effect and predation or infection can turn continuous-front invasion into a patchy invasion. This is a generic property and can be observed in continuous systems as well as in discrete ones. Theoretical results are consistent with (some) field data These results seem to lead to a new paradigm of invasion pattern: patchy spread instead of continuous front The patchy invasion provides a mechanism of invasion at the edge of extinction

53 Conclusions Interplay between the Allee effect and predation or infection can turn continuous-front invasion into a patchy invasion. This is a generic property and can be observed in continuous systems as well as in discrete ones. Theoretical results are consistent with (some) field data These results seem to lead to a new paradigm of invasion pattern: patchy spread instead of continuous front The patchy invasion provides a mechanism of invasion at the edge of extinction Invasion success depends on increase in predation or infection strength in a non-monotonous way

54 Open questions & challenges Mathematical aspects: a qualitative theory of the patchy invasion is largely missing! Evolutionary aspects: what are the mechanisms that would have led to such an evolutionary response? Environmental aspects: There have been more and more cases of patchy invasion discovered over the recent years. Is this a result of the improved data collection techniques or does it reflect the changes in the inherent biological dynamics, e.g. due to the climate change?

55 Acknowledgements Andrew Morozov (Leicester, UK) Masha Jancovic (Leicester, UK) Diomar Mistro (Santa Maria, Brazil) Luiz Rodrigues (Santa Maria, Brazil)