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# The Mathematics Behind Biological Invasions

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## Abstract

This book investigates the mathematical analysis of biological invasions. Unlike purely qualitative treatments of ecology, it draws on mathematical theory and methods, equipping the reader with sharp tools and rigorous methodology. Subjects include invasion dynamics, species interactions, population spread, long-distance dispersal, stochastic effects, risk analysis, and optimal responses to invaders. While based on the theory of dynamical systems, including partial differential equations and integrodifference equations, the book also draws on information theory, machine learning, Monte Carlo methods, optimal control, statistics, and stochastic processes. Applications to real biological invasions are included throughout. Ultimately, the book imparts a powerful principle: that by bringing ecology and mathematics together, researchers can uncover new understanding of, and effective response strategies to, biological invasions. It is suitable for graduate students and established researchers in mathematical ecology.
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## Chapters (9)

We provide an overview of mathematical tools and approaches for modeling the population dynamics of invasive species. We begin with a brief qualitative analysis of biological invasion to show how it can be split into different stages, each stage having its own determinants in terms of invasive species management. We then introduce the modeling framework that will be used in the rest of the book. A detailed description of relevant nonspatial population models is followed by a discussion of the spatial dynamics. We will show that a variety of mathematical techniques may be used to describe different aspects of invasion dynamics at different invasion stages.
We revisit the baseline model of biological invasion consisting of a single partial differential equation of reaction–diffusion type. In spite of being one of the oldest models of biological invasion, it remains a valid and useful tool for understanding the spatiotemporal population dynamics of invasive species. We first apply this model to alien species establishment and show how to decide whether an initial population distribution results in extinction or survival. We then use the model to reveal the properties characterizing invasive species spread.
We consider how the rate and pattern of spread of an invasive (alien) species can be affected by interactions with other species, e.g., species in the native community or biological control agents. We show that interspecific interactions can decrease the rate of spread significantly or can stop the propagating invasion front completely, and may even reverse it, hence resulting in the failure of the invasion and the eradication of the alien species. We also show that interspecific interactions can change the pattern of spread by turning the propagating front into patchy spread.
Long-distance dispersal is a key factor driving rapid spread of invasive species. In this chapter, we develop models for dispersal kernels, which describe the spatial distribution of propagules relative to their parent. Dynamical systems that couple dispersal kernels with population growth can be formulated in either discrete time (integrodifference equations) or continuous time (integrodifferential equations). We show that, when long-distance dispersal via the dispersal kernel is incorporated into a dynamical system, the population can spread an order of magnitude faster than predicted by the equivalent reaction–diffusion models of the sort seen in Chap. 3 Indeed, when dispersal kernels are fat-tailed, the population spread can actually accelerate continually, leading to theoretically infinite asymptotic spreading speeds. We show how the concept of rapid invasive spread has applications not only to invasive species but also to the spread of infectious disease. The idea of an accelerating invasion can also be understood via the concept of stratified diffusion, where a spatially implicit model is used to keep track of the size distribution of localized invasion processes. Finally, the results regarding rapid spread implicitly assume that there are no Allee effects. We show how Allee effects obstruct rapid and accelerating invasions, bringing the spreading speed down to much lower levels.
This chapter focuses on connecting integrodifference models to biological data, so theoreticians can readily calculate spreading speeds for an invader based on available data. We investigate the use of a nonparametric estimator, which avoids the need to specify a functional form for the dispersal kernel. This approach is extended to include a histogram estimator, which can be applied to the case where data are binned into distance classes. We show how calculations differ slightly for different types of one-dimensional data (radial dispersal distances versus linear, one-dimensional dispersal distances), and we provide explicit formulae for each case. In some situations, dispersal distances come not from data but from complex computer simulations. In this case, Monte Carlo simulations can provide data for the nonparametric estimator, which in turn yields a straightforward estimate for spreading speed. Finally, the complexity of stage structure can be included in the integrodifference equation, yielding spreading speeds for stage-structured integrodifference models. Applications of the theory in this chapter are made to spreading speeds for Drosophila and teasel.
Uncertainty is a hallmark of early invasion processes. Mathematical descriptions of this uncertainty can help us assign probabilities to possible invasion outcomes. This chapter starts with a hierarchical model of invasion, describing the process of transport, introduction, and survival to reproduction. The model yields a probability of successful establishment for potential invaders, as well as the distribution of times needed before a successful invasion will occur. An extension of the model includes the possibility of an invasion bottleneck produced by the need for sexual reproduction. Environmental variability has a role to play in invasion success. This aspect is investigated using classical discrete- and continuous-time models for population growth under stochasticity. Here, Jensen’s inequality is applied to show that an invasion taking place amid discrete-time random environmental fluctuations may not succeed, even if it would succeed in a constant environment. Finally, a general, but approximate, method for understanding the impacts of various types of uncertainty (environmental, demographic, and immigration) on invasion success is formulated using stochastic differential equations. This method is then used to model invasion success for populations with an Allee effect. We apply the theory in this chapter to understand the invasive outcomes for aquatic invasive species, such as the Chinese mitten crab and the apple snail.
We investigate variability in the spread of invasive species. The models are stochastic spatiotemporal processes, describing the density of invasive species in space and time. Our focus is on the rate at which such processes spread spatially. We first examine the effects of environmental stochasticity on spatial spread by means of stochastic integrodifference and reaction–diffusion models. Here, we analyze both the wave solution for the expected density of individuals and the wave solution for a given realization of the stochastic process, as well as the variability that this can exhibit. Then we turn our analysis to the effects of demographic stochasticity on spatial spread. This can be described by its effects on the expected density and also by its effects on the velocity of the furthest-forward location for the population. Finally, we consider nonlinear stochastic models for patchy spread of invasive species, showing how patchiness in the leading edge of an invasion process can dramatically slow the invasive spreading speed. We apply the theory in this chapter to understand mathematically the rate of recolonization of trees in North America after the last ice age.
What are the chances that a species will become invasive in a given area? This is the core question of invasion risk. In this chapter, we examine the three key aspects that affect the outcome of this question: propagule pressure, species traits, and geographical traits. As well as employing traditional mathematical tools, we will use methods from machine learning, a branch of computer science, to attack this problem. We demonstrate the fundamental importance of human interactions in assessing risk, and how to quantify the economic consequences of human decisions. Finally, we reveal how mathematics can teach important lessons about political short-termism.
Invasive species impact human activities. We can respond using a range of strategies, including detection, adaptation, and control (via prevention and eradication). But what strategies are best, and when should we use them? To answer these questions, we couple models for species establishment and spread with those for impact and control. A final level of modeling puts associated costs and benefits into an economic framework. Optimal strategies are then based on a cost–benefit analysis that involves methods of optimization, including optimal control theory, stochastic dynamic programming, and linear programming.
... In a rather general case, the fraction of habitat lost P is expected to follow a power-law [211,215]: Fig. 5. Per capita growth rate in the baseline single-species model (11) for different strengths of the Allee effect: curve 1: no Allee effect, curves 2 and 3: weak Allee effect, curve 4: strong Allee effect. From [170]. ...
... In the second case, f (u) is not monotonous: it increases at small u and decreasing for large u, such that the maximum per capita growth rate is reached at some positive value of the population density. The former case is often referred to as logistic growth (sometimes as 'generalized logistic growth') and the latter case as population dynamics with an Allee effect [170,205]. The Allee effect is increasingly identified as innate in many species [69,70,284] which suggests that population growth with the Allee effect is more relevant for the general model (11) than logistic growth. ...
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... While the most common approach of single species reaction-diffusion growth in biology is via the logistic growth (Lewis et al. 2016;Tian and Ruan 2018), the shape parameter v of the generalised logistic function was found to be near zero, which corresponds to Gompertz population growth. The Fisher equation with this growth term is rarely found in the population dynamics literature, as in Kandler and Unger (2010), and to our knowledge, it has not been used in specific studies of mosquito population dynamics, both theoretical and field. ...
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