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Modelling the dynamics of invasion and control of competing green crab genotypes

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Establishment of invasive species is a worldwide problem. In many jurisdictions, management strategies are being developed in an attempt to reduce the environmental and economic harm these species may cause in the receiving ecosystem. Scientific studies to improve understanding of the mechanisms behind invasive species population growth and spread are key components in the development of control methods. The work presented herein is motivated by the case of the European green crab (Carcinus maenas L.), a remarkably adaptable organism that has invaded marine coastal waters around the globe. Two genotypes of European green crab have independently invaded the Atlantic coast of Canada. One genotype invaded the mid-Atlantic coast of the USA by 1817, subsequently spreading northward through New England and reaching Atlantic Canada by 1951. A second genotype, originating from the northern limit of the green crabs European range, invaded the Atlantic coast of Nova Scotia in the 1980s and is spreading southward from the Canadian Maritime provinces. We developed an integrodifference equation model for green crab population growth, competition and spread, and demonstrate that it yields appropriate spread rates for the two genotypes, based on historical data. Analysis of our model indicates that while harvesting efforts have the benefit of reducing green crab density and slowing the spread rate of the two genotypes, elimination of the green crab is virtually impossible with harvesting alone. Accordingly, a green crab fishery would be sustainable. We also demonstrate that with harvesting and restocking, the competitive imbalance between the Northern and Southern green crab genotypes can be reversed. That is, a competitively inferior species can be used to control a competitively superior one.

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The authors are grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) and MITACS for project funding. Also, a special thank you to C. McCarthy, Kejimkujik National Park, for sharing unpublished results of green crab control experiments.

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Correspondence to Lisa Kanary.


Appendix A: Stability of the positive steady state

We prove that the unique positive steady state of the non-spatial model equation (1) is stable whenever it exists. The single-genotype non-spatial model is given by

$$ N_{t+1} \,=\,F(N_t) = \left(\frac{P_{m}}{1+\alpha_P N_t} + \frac{S_{m}}{1+\alpha_S N_t} \frac{R_{m}}{1+\alpha_R N_t} \right) N_t. $$

Model (32) possess the trivial equilibrium, and has a unique positive equilibrium provided P m + S m R m > 1 and at least one of α S , α P , α R is positive. The unique positive equilibrium satisfies the equation

$$ 1 - \frac{P_{m}}{1 + \alpha_p N^{*}} = \frac{S_{m}}{1+\alpha_S N^{*}} \frac{R_{m}}{1+\alpha_R N^{*}}. $$

It is stable when \(|F^{\prime }(N^{*})|<1\). Substituting, we calculate

$$\begin{array}{@{}rcl@{}} F^{\prime}(N^{*})&=& \frac{P_{m}}{(1 + \alpha_p N^{*})^2} + \frac{R_{m}\,S_{m}(1+\alpha_sN^{*})(1+\alpha_rN^{*}) - R_{m}\,S_{m}N^{*}(\alpha_s(1 + \alpha_rN^{*}) + \alpha_r(1 + \alpha_sN^{*}))}{(1 + \alpha_sN^{*})^2(1 + \alpha_r N^{*})^2}. \end{array} $$

which can be simplified by using Eq. 33 to

$$\begin{array}{@{}rcl@{}}F^{\prime}(N^{*}) &=& \frac{P_{m}}{(1 + \alpha_p N^{*})^2} + \left(1 - \frac{P_{m}}{1 + \alpha_p N^{*}}\right)\\ &&\times\left(1 - \frac{N^{*}\alpha_s}{1 + \alpha_sN^{*}} - \frac{N^{*}\alpha_r}{1 + \alpha_rN^{*}}\right). \end{array} $$

Since \(\frac {P_{m}}{1 + \alpha _p N^{*}} < 1,\) the first bracketed quantity in Eq. 35 is strictly positive and smaller than unity. The second bracketed quantity in Eq. 35 is between negative one and positive one. Therefore, \(|F^{\prime }(N^{*})| \leq 1,\) and the unique positive equilibrium is always stable. Note that, solutions near N may oscillate since the updating function in Eq. 32 is, in general, not monotone. However, it turns out that for our point-estimates listed in Table 1, the updating function in Eq. 32 is monotone.

Appendix B: Monotonicity and the joint spreading speed

We show that a change of variables turns the competitive system (7) into a cooperative system, and we discuss the similarities and differences with the work in Weinberger et al. (2002, 2007).

We can write the non-spatial competitive system as

$$\begin{array}{@{}rcl@{}} N_{t+1} &=& f(E_t) N_t \\ M_{t+1} &=& f(\rho E_t)M_t, \end{array} $$

where E t = N t + κ M t . If updating function Nf(N)N is monotone on [0,M ], then the domain [0,M ]2 is invariant for this system. While our updating function need not be monotone, it is monotone for the parameter ranges we found in the literature. The new variables u t = N t , and v t = M M t . satisfy the equations

$$\begin{array}{@{}rcl@{}} u_{t+1} &=& f(u_t + \kappa(M^{*} - v_t)) u_t := \mathcal{F}(u_t, v_t) \\ v_{t+1} &=& f(u_t + \kappa(M^{*} - v_t)) \\ &&+ M^{*}(1 - f(u_t + \kappa(M^{*} - v_t))) := \mathcal{G}(u_t, v_t). \end{array} $$

We verify that system (37) is cooperative. By definition (Smith 1995), system (37) is cooperative if

$$\frac{\partial \mathcal{F}}{\partial v} \geq 0, \qquad \frac{\partial \mathcal{G}}{\partial u} \geq 0.$$

Since \(f^{\prime }(E) < 0,\) we have

$$\begin{array}{@{}rcl@{}} \frac{\partial \mathcal{F}}{\partial v} &=& f^{\prime}(E)(-\kappa) \geq 0 \\\notag \frac{\partial \mathcal{G}}{\partial u} &=& f^{\prime}(E)(v - M^{*}) \geq 0, \ \text{if} \ v < M^{*}. \end{array} $$

Thus, we see that system (37) is cooperative in the biologically realistic region 0 ≤ uN , 0 ≤ vM .

The above results have implications for the spreading speed of the spatial system in Eq. 10. Weinberger et al. (2002, 2007) gave sufficient conditions under which the joint spreading speed of the two genotypes in a special case of system (10) exists, is linearly determinate and is given by the speed of the dominant invader. Namely, for P m = 0 and either α S = 1 and α R = 0 or vice-versa, system (10) is the one studied by Weinberger et al. (2002, 2007). The crucial part of the proof requires that the system can be transformed into a monotone system. In the general case, i.e. P m > 0 and α s α R > 0 the situation is unclear. For one, the operator is not compact. Secondly, the operator may not be monotone. Our numerical results suggest that the conclusions still hold, but the mathematical proof is a future challenge.

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Kanary, L., Musgrave, J., Tyson, R.C. et al. Modelling the dynamics of invasion and control of competing green crab genotypes. Theor Ecol 7, 391–406 (2014).

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