Elsevier

Mathematical Biosciences

Volume 215, Issue 2, October 2008, Pages 158-166
Mathematical Biosciences

Biological invasions: Deriving the regions at risk from partial measurements

https://doi.org/10.1016/j.mbs.2008.07.004Get rights and content

Abstract

We consider the problem of forecasting the regions at higher risk for newly introduced invasive species. Favourable and unfavourable regions may indeed not be known a priori, especially for exotic species whose hosts in native range and newly-colonised areas can be different. Assuming that the species is modelled by a logistic-like reaction–diffusion equation, we prove that the spatial arrangement of the favourable and unfavourable regions can theoretically be determined using only partial measurements of the population density: (1) a local ‘spatio-temporal’ measurement, during a short time period and, (2) a ‘spatial’ measurement in the whole region susceptible to colonisation. We then present a stochastic algorithm which is proved analytically, and then on several numerical examples, to be effective in deriving these regions.

Introduction

Because of trade globalisation, a substantial increase in biological invasions has been observed over the last decades (e.g. Liebhold et al. [1]). These invasive species are, by definition [2], likely to cause economic or environmental harm or harm to human health. Thus, it is a major concern to forecast, at the beginning of an invasion, the areas which will be more or less infested by the species.

Because of their exotic nature, invading species generally face little competition or predation. They are therefore well adapted to modelling via single-species models.

Reaction–diffusion models have proved themselves to give good qualitative results regarding biological invasions (see the pioneering paper of Skellam [3], and the books [4], [5] and [6] for review).

The most widely used single-species reaction–diffusion model, in homogeneous environments, is probably the Fisher–Kolmogorov [7], [8] model:ut=DΔu+u(μ-γu),t>0,xΩRN,where u=u(t,x) is the population density at time t and space position x, D is the diffusion coefficient, μ corresponds to the constant intrinsic growth rate, and μγ is the environment’s carrying capacity. Thus γ measures the susceptibility to crowding effects.

On the other hand, the environment is generally far from being homogeneous. The spreading speed of the invasion, as well as the final equilibrium attained by the population are in fact often highly dependent on these heterogeneities [4], [9], [10], [11]. A natural extension of (1.1) to heterogeneous environments has been introduced by Shigesada, Kawasaki, Teramoto [12]:ut=(D(x)u)+u(μ(x)-γ(x)u),t>0,xΩRN.In this case, the diffusivity matrix D(x), and the coefficients μ(x) and γ(x) depend on the space variable x, and can therefore include some effects of environmental heterogeneity.

In this paper, we consider the simpler case where D(x) is assumed to be constant and isotropic and γ is also assumed to be positive and constant:ut=DΔu+u(μ(x)-γu),t>0,xΩRN.The regions where μ is high correspond to favourable regions (high intrinsic growth rate and high environment carrying capacity), whereas the regions with low values of μ are less favourable, or even unfavourable when μ<0. In what follows, in order to obtain clearer biological interpretations of our results, we say that μ is a ‘habitat configuration’.

With this type of model, many qualitative results have been established, especially regarding the influence of spatial heterogeneities of the environment on population persistence, and on the value of the equilibrium population density [4], [9], [13], [14], [15]. However, for a newly introduced species, like an invasive species at the beginning of its introduction, the regions where μ is high or low may not be known a priori, particularly when the environment is very different from that of the species native range.

In this paper, we propose a method of deriving the habitat configuration μ, basing ourselves only on partial measurements of the population density at the beginning of the invasion process. In Section 2, we begin by giving a precise mathematical formulation of our estimation problem. We then describe our main mathematical results, and we link them with ecological interpretations. These theoretical results form the basis of an algorithm that we propose, in Section 3, for recovering the habitat configuration μ. In Section 4, we provide numerical examples illustrating our results. These results are further discussed in Section 5.

Section snippets

Model and hypotheses

We assume that the population density uγ is governed by the following parabolic equation:tuγ=DΔuγ+uγ(μ(x)-γuγ),t>0,xΩ,uγ(t,x)=0,t>0,xΩ,uγ(0,x)=ui(x)inΩ,(Pμ,γ)where Ω is a bounded subdomain of Rd with boundary Ω. We will denote Q:=(0,+)×Ω and Σ:=(0,+)×Ω.

The growth rate function μ is a priori assumed to be bounded, and to take a known constant value outside a fixed compact subset Ω1 of Ω:μM:={ρL(Ω),-MρMa.e., andρminΩΩ1},for some constants m,M, with M>0; the notation ‘a.e.’ means

Simulated annealing algorithm

Let (t0,t1) be a fixed time interval, and ωΩ1 be fixed. We assume that we have measurements of the solution uγ(t,x) of (Pμ,γ) over (t0,t1)×ω, and of uγ(t0+t12,x) in Ω. However, the function μ and the constant γ are assumed to be unknown. Our objective is to build an algorithm for recovering μ.

Remark 3.1

When the function uγ is known, the computation of Gμ(γ,·) does not require the knowledge of γ.

The function μ is assumed to belong to a known finite subset E of M, equipped with a neighbourhood system. We

Numerical computations

In this section, in one-dimensional and two-dimensional cases, we check that the algorithm presented in Section 3 can work in practice.

In each of the four following examples, we fixed the sets Ω, Ω1 and M and we defined a finite subset EM equipped with a neighbourhood system. Then, for a fixed habitat configuration μE we computed, using a second-order finite elements method, the solution u(t,x) of (Pμ,γ), for D=1, γ=0.1, t(0,0.5), and for a given initial population density ui. Then, we fixed

Discussion and conclusion

We have shown that, for an invasive species whose density is well modelled by a reaction–diffusion equation, the spatial arrangement of the favourable and unfavourable regions can be measured indirectly through the population density at the beginning of the invasion. More precisely, we considered a logistic-like reaction–diffusion model, and we placed ourselves under the assumption that the initial population density was far from the environment carrying capacity (it can be reasonably assumed

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments and insightful suggestions. This study was partly supported by the French ‘Agence Nationale de la Recherche’ within the project URTICLIM ‘Anticipation des effets du changement climatique sur l’impact écologique et sanitaire d’insectes forestiers urticants’ and by the European Union within the FP 6 Integrated Project ALARM- Assessing LArge-scale environmental Risks for biodiversity with tested Methods

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