## Abstract

Invasive species have been recognized as a leading threat to biodiversity. In particular, lakes are especially affected by species invasions because they are closed systems sensitive to disruption. Accurately controlling the spread of invasive species requires solving a complex spatial-dynamic optimization problem. In this work we propose a novel framework for determining the optimal management strategy to maximize the value of a lake system net of damages from invasive species, including an endogenous diffusion mechanism for the spread of invasive species through boaters’ trips between lakes. The proposed method includes a combined global iterative process which determines the optimal number of trips to each lake in each season and the spatial-dynamic optimal boat ramp fee.

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## Notes

- 1.
Note that this utility function can be further generalized to allow for nonlinear impacts of income and to allow for congestion to impact boaters (i.e. independence among boaters’ utilities).

- 2.
We assume that the invasion status is updated at the end of the season such that boaters’ trip decisions depend on \(x_{s-1}\) and at the end of the season when the invasion status is updated \(x_s\) depends on the boating decisions in season

*s*. - 3.
We take \(a=2.824153\) which approximates the sigmoid with \(\ell _\infty \) error at most 0.056075.

- 4.
Accounting for a non-constant welfare loss that depended on the number of lakes invaded was found to be minor in Zipp et al. (2019) (around 2%), however, future work could allow the welfare loss per boater to depend on the number of invaded lakes.

- 5.
Clearly, allowing a negative number of boating trips would not be realistic.

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### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of Katherine Y. Zipp was partially supported by the Department of Agricultural Economics, Sociology, and Education at Penn State, the USDA National Institute of Food and Agriculture and Multistate Hatch Appropriations under Project # PEN04631 and Accession # 1014400, and a seed grant from the Institute for CyberScience at Penn State. The work of Yangqingxiang Wu was partially supported by the Department of Agricultural Economics, Sociology, and Education at Penn State. The work of Ludmil T. Zikatanov was partially supported by NSF Grants DMS-1720114 and DMS–1819157 and a seed grant from the Institute for CyberScience at Penn State.

## Appendix: On the convergence of the algorithm

### Appendix: On the convergence of the algorithm

The algorithm we proposed above utilizes a sequence of quadratic programming problems which, as we have shown in Sect. 2.5, are solvable analytically. This is a novel approach and to support this design we give a brief analysis of its convergence.

We note that the goal is to maximize \({{\mathscr {G}}}(\varvec{b})\) given by [(see (11)]:

In the equations above, one can think of \({{\mathscr {F}}}(\widetilde{\varvec{b}},\varvec{b})\) as extending \({{\mathscr {G}}}\) from the “line” \(\varvec{b}=\widetilde{\varvec{b}}\) to the “plane” \((\widetilde{\varvec{b}},\varvec{b})\). It should be clear that we use the terms “line” and “plane” loosely here to identify the multidimensional analogues of such objects.

Since we are free to choose the extension \({{\mathscr {F}}}\) we may assume that we have extended the profit function so that

Recall that, Algorithm 3, for a given \(\varvec{b}_k\) maximizes \({{\mathscr {F}}}(\varvec{b}_k,\varvec{c})\) with respect to \(\varvec{c}\), and the optimal value of \({{\mathscr {F}}}\) is at \(\varvec{c} = \varvec{b}_{k+1}\). Note that this implies that

where \(\varvec{b}_*\) is the optimal solution which maximizes \({{\mathscr {F}}}(\varvec{b},\varvec{b})\) (the optimal value we want to find). As we have shown, such relation holds because at \(\varvec{c}=\varvec{b}_{k+1}\), the function \({{\mathscr {F}}}(\varvec{b}_k,\varvec{c})\) viewed as function of \(\varvec{c}\), is at a maximum. Therefore, the value of \({{\mathscr {F}}}(\varvec{b}_k,\varvec{b}_{k+1})\) cannot be smaller than the value of \({{\mathscr {F}}}(\varvec{b}_k,\varvec{b}_*)\).

If we further assume that \(\lim _{k\rightarrow \infty }\varvec{b}_k=\varvec{b}_{\infty }\) (which we found numerically to be always true in all examples we tried) and take the limit on both sides. Since \({{\mathscr {F}}}\) is continuous (not necessarily differentiable, just merely continuous is enough here), we obtain that

The first inequality holds because \(\varvec{b}_*\) is the optimal solution, the second holds because of the limit w.r.t *k* in (17) and last inequality follows from (16).

Since the left and right sides of these inequalities are equal, we must have equality everywhere. In conclusion, under the simple assumptions we made above, if the sequence of iterates converges then it converges to an optimal value of the objective function.

To make the argument precise, let us point out that the sequence of *all* iterates may not converge, but may have one, two or more convergent subsequences. As is known, by Heine–Borel theorem, as long as this sequence is bounded (which it is because of the constraints), it must be a convergent subsequence. The considerations given above apply to any convergent subsequence as well. We can then conclude that the function values of the limit of any such convergent subsequence is the optimal value of the benefit function. We have the following result and its proof is an immediate consequence of the considerations above.

### Lemma 1

If the extension satisfies (16), then for every convergent subsequence of iterates \(\{b_{k_j}\}_{j=1}^J\), \(\lim \limits _{j\rightarrow \infty } b_{k_j}=b_{\infty ,j}\) the function values converge to the optimal value, namely,

We further remark that, while the optimal solutions, i.e. the limits of subsequences, \(\{b_{\infty ,j}\}_{j=1}^J\) may be different, the function values at such points are the same.

Finally, let us note that the technique of extending a function to a higher dimensional space (from one variable to two) is well known in the theory of partial differential equations and can be viewed as a special *reguralization*. The reason is that a less regular problem can be extended to more regular and better behaved in higher dimension.

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### Cite this article

Zipp, K.Y., Wu, Y., Wu, K. *et al.* Optimal spatial-dynamic management to minimize the damages caused by aquatic invasive species.
*Lett Spat Resour Sci* **12, **199–213 (2019). https://doi.org/10.1007/s12076-019-00237-x

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### Keywords

- Invasive species
- Spatial-dynamic management
- Convex optimization
- Bioeconomic

### JEL Classification

- Q20
- Q50
- Q57