Biological invasions: Deriving the regions at risk from partial measurements
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:where is the population density at time t and space position x, D is the diffusion coefficient, corresponds to the constant intrinsic growth rate, and is the environment’s carrying capacity. Thus 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]:In this case, the diffusivity matrix , and the coefficients and depend on the space variable x, and can therefore include some effects of environmental heterogeneity.
In this paper, we consider the simpler case where is assumed to be constant and isotropic and is also assumed to be positive and constant:The regions where is high correspond to favourable regions (high intrinsic growth rate and high environment carrying capacity), whereas the regions with low values of are less favourable, or even unfavourable when . In what follows, in order to obtain clearer biological interpretations of our results, we say that 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 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 , 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 . 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 is governed by the following parabolic equation:where is a bounded subdomain of with boundary . We will denote and .
The growth rate function is a priori assumed to be bounded, and to take a known constant value outside a fixed compact subset of :for some constants , with ; the notation ‘a.e.’ means
Simulated annealing algorithm
Let be a fixed time interval, and be fixed. We assume that we have measurements of the solution of over , and of in . However, the function and the constant are assumed to be unknown. Our objective is to build an algorithm for recovering . Remark 3.1 When the function is known, the computation of does not require the knowledge of .
The function is assumed to belong to a known finite subset E of , 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 , and and we defined a finite subset equipped with a neighbourhood system. Then, for a fixed habitat configuration we computed, using a second-order finite elements method, the solution of , for , , , and for a given initial population density . 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
References (25)
- et al.
Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts
J. Math. Pures Appl.
(2005) - et al.
Traveling periodic waves in heterogeneous environments
Theor. Popul. Biol.
(1986) - et al.
Mathematical analysis of the optimal habitat configurations for species persistence
Math. Biosci.
(2007) - et al.
Invasion by exotic forest pests: a threat to forest ecosystems
For. Sci. Monogr.
(1995) - National Invasive Species Information Center, USDA 2006, Executive Order...
Random dispersal in theoretical populations
Biometrika
(1951)- et al.
Biological Invasions: Theory and Practice
(1997) Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants
(1998)- et al.
Diffusion and Ecological Problems – Modern Perspectives
(2002) The wave of advance of advantageous genes
Ann. Eugenics
(1937)
Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique
Bull. Univ. État Moscou, Série Int. A
Analysis of the periodically fragmented environment model. I. Species persistence
J. Math. Biol.
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