A model for the spread of invasive species that brings the landscape into the equation

An invasive species can be any kind of living organism—an amphibian (like the cane toad), plant, insect, fish, fungus, bacteria, or even an organism’s seeds or eggs—that is not native to an ecosystem and causes harm. They can harm the environment, the economy, or even human health. Species that grow and reproduce quickly, and spread aggressively, with potential to cause harm, are given the label “invasive.” An invasive species does not have to come from another country, just from another ecosystem.

Invasive species are primarily introduced by human activities, often unintentionally. People, and the goods they use, travel around the world very quickly, and they often carry uninvited species with them. Ships can carry aquatic organisms in their ballast water, while smaller boats may carry them on their propellers. Insects can get into wood, shipping palettes, and crates that are shipped around the world. Some ornamental plants can escape into the wild and become invasive. And some invasive species are intentionally or accidentally released pets.

The point is that invasive species are among the leading threats to native wildlife. Approximately 42 percent of threatened or endangered species are at risk due to invasive species. ¹

Human health and economies are also at risk from invasive species. The impacts of invasive species on natural ecosystems and economy cost billions of dollars each year.

Once introduced, the management of the spread of the species is a challenging problem due to the uncertainty surrounding its characteristics and the usually limited resources allocated to its control. Actually, measures to protect the native species of a region from invasive species are costly for local governments. On the other hand, the management often starts long after the appearance of the invasive species, when the ecological damage is already visible and the species well-established, taking an additional financial toll.

An important consideration is the need to account for the temporal dynamics of the invader population. Managers have to optimize their choices not only for the present, but must also take into account possible future scenarios. The introduction of the cane toad to Australia from Hawaii in 1935 in an attempt to control the native grey-backed cane beetle and its subsequent, uncontrolled, spread throughout the continent is a cautionary tale in this respect.

If the understanding of population persistence and competitive outcomes can provide long-term information, then short-term predictions of spread are often more important in the management of invasive species as fast responses can potentially save public administrations sizeable amounts of money. Optimal management strategies generally rely on the coupling of biological models with optimization procedures. Within such a framework, the modeling of invasive species and the dynamics of their dispersal can be of great help to environment conservation agents and policymakers.

Relying on the assumption that the dispersal of an individual is random, but the density of individuals at the scale of the population can be considered smooth, reaction-diffusion models are a good trade-off between model complexity and flexibility for use in different situations. Reaction–diffusion systems are mathematical models which correspond to several physical phenomena: the most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. In the case of invasive species we find a similar pattern, a local introduction and reproduction, and the spread due to migration (animals) or dipersal of seeds (plants). Hence, several scientists have applied continuous reaction-diffusion models to these problems.

Recently, a reaction-diffusion model was applied to simulate the effect of management efforts on the distribution of Burmese pythons in the Everglades (Florida, USA). The reaction-diffusion method was shown to yield better prediction accuracy than others. But there was an important limitation of the study, namely, that the region of interest was treated as an isolated environment with zero flux boundary conditions imposed. This is a common feature to many works (not only in reaction-diffusion models), where the considered environment is either an island or a region bounded by impenetrable borders and where the effect of migration from neighboring domains is not considered.

Now, a team of researchers has developed a continuous, time-dependent, nonlinear reaction-diffusion model ² for the species density that arises from a generalization of the Fisher’s equation, one of the simplest semilinear reaction-difussion equations.

A major feature of this work is that landscape heterogeneities are accounted for by including in the computational domain the significant geographical features of the area by acting directly on the coefficients of the model that are both spatially and temporally dependent. In particular, the researchers show that the model can account for elevation in a natural way.

Elevation being a major trait in landscape heterogeneity, pretty much overlooked in the literature, despite the fact that quite often species’ habitats are bounded by it. This is now more important than ever due to climate change. Recent studies have found that increased temperature and changes in humidity have rendered higher altitudes suddenly hospitable for new species that might pose a threat to indigenous ones. From this perspective, the simplicity with which the new model is able to account for elevation can become a major asset.

The reaction-diffusion model is numerically approximated by the finite element method, which is able to treat arbitrarily shaped boundaries, like the ones of a geographical region. Finally, finite elements can easily be employed on adaptive grids that feature finer resolution in specific regions of interest, for instance co-localized with peculiar landscape characteristics.

As a proof of concept, the researchers demonstrate the practical applicability of the proposed method by quantifying the uncertainty in the spread of a generic invasive population in the Basque Country area in northern Spain.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may be copied verbatim or almost verbatim from the referenced research paper.