Identifying non-invasible habitats for marine copepods using temperature-dependent <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>

Harshana Rajakaruna; Carly Strasser; Mark Lewis

doi:10.1007/s10530-011-0104-x

Biological Invasions

March 2012, Volume 14, Issue 3, pp 633–647 | Cite as

Identifying non-invasible habitats for marine copepods using temperature-dependent R ₀

Authors
Authors and affiliations

Harshana RajakarunaEmail author
Carly Strasser
Mark Lewis

Original Paper

First Online: 23 September 2011

Received: 28 October 2010

Accepted: 09 September 2011

Abstract

If a non-indigenous species is to thrive and become invasive it must first persist under its new set of environmental conditions. Net reproductive rate (R ₀) represents the average number of female offspring produced by a female over its lifetime, and has been used as a metric of population persistence. We modeled R ₀ as a function of ambient water temperature (T) for the invasive marine calanoid copepod Pseudodiaptomus marinus, which is introduced to west coast of North America from East Asia by ship ballast water. The model was based on temperature-dependent stage-structured population dynamics given by a system of ordinary differential equations. We proposed a methodology to identify habitats that are non-invasible for P. marinus using the threshold of R ₀(T) < 1 in order to identify potentially invasible habitats. We parameterized the model using published data on P. marinus and applied R ₀(T) to identify the range of non-invasible habitats in a global scale based on sea surface temperature data. The model predictions matched the field evidence of species occurrences well.

Keywords

Net reproductive rate Invasive species Marine copepods Pseudodiaptomus marinus Temperature Stage-structured population models Ordinary differential equations Ecological modeling Habitat invasibility Habitat suitability

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Notes

Acknowledgments

Financial support for HR and CS came from the NSERC-funded Canadian Aquatic Species Network (CAISN). HR also acknowledges the Department of Biological Sciences, University of Alberta for providing financial support. MAL gratefully acknowledges an NSERC Discovery Grant and a Canada Research Chair. The authors thank Alex Potapov at the Centre for Mathematical Biology, University of Alberta, and Claudio DiBacco at the Bedford Institute of Oceanography, Halifax, for valuable suggestions.

Appendix 1: Deriving R ₀ from graph theoretic method

Following the method given in de-Camino-Beck and Lewis (2008), here we have a real 12 × 12 matrix $ (F\lambda^{ - 1} - V) = a_{ij} $ after decomposing matrix A from Eq. 1 into matrices F, fertility, and V, transition. Hence, for matrix (Fλ⁻¹ − V), there corresponds a labeled directed graph, D(Fλ⁻¹ − V), with nodes 1, 2, …, 12, and a directed edge (arc) j → i. The weight of this arc is a _ij, and D(Fλ⁻¹ − V) has a loop at node i of weight a _ij if a _ij ≠ 0. Thus, we can draw the diagraph, D(Fλ⁻¹ − V), as follows.

We created trivial nodes using graph reduction Rule 1 in de-Camino-Beck and Lewis (2008) by reducing the loops −a _ii < 0 to −1 at node i’s, for every arc entering i divided by weight a _ii. Thus the diagraph will be reduced to the following.

Using Rule 2 in de-Camino-Beck and Lewis (2008), by eliminating arcs through trivial nodes, here we replaced two arcs at a time by j → k with weights equal to the product of weights on arc j → i and i → k, for trivial nodes i on a path j → i → k. Thus, it finally yields the following diagraph with a single node.

Finally, we set the weight of this loop to zero giving and equation of lambda. The smallest positive roots of this equation yielded R ₀.

$$ R_{0} = \frac{q\beta }{{\mu_{12} }}\prod\limits_{i = 1}^{11} {\left( {\frac{{\gamma_{i} }}{{\gamma_{i} + \mu_{i} }}} \right)} $$

Furthermore, when there are 2 sub stages in each stage (that is k = 2), the initial graph is given as follows:

Using Rule 1, this can be reduced as follows.

It finally yields,

Thus, $ R_{0} = \frac{q\beta }{{\mu_{12} }}\prod\nolimits_{i = 1}^{11} {\left( {\frac{{\gamma_{i} }}{{\gamma_{i} + \mu_{i} }}} \right)^{2} } . $

Similarly, for any k sub stages, it yields, $ R_{0} = \frac{q\beta }{{\mu_{12} }}\prod\nolimits_{i = 1}^{11} {\left( {\frac{{\gamma_{i} }}{{\gamma_{i} + \mu_{i} }}} \right)^{k} } $.

The same result can be easily derived from R ₀ = ρ[FV ⁻¹] also.

Appendix 2: General solution for n _a(t)

We obtained the following general solution for n _a(t), the proportion of individuals in a given stage a at time t in Eq. 1:

$$ \begin{array}{*{20}c} {n_{a} (t) = \left( {\prod\limits_{i = 1}^{a - 1} {\gamma_{i} } } \right)\left( {\underline{b}_{a} .\underline{\nu }_{a} } \right)} & {{\text{for}}\,a > 1;} \\ {n_{1} (t) = e^{{ - \sigma_{1} t}} \gamma_{1} } & { {\text{for}}\,a = 1;} \\ \end{array} $$

(5)

where, σ_i = (γ_i + μ_i) such that γ_i > 0 and μ_i > 0 for any stage i and σ_ij = (σ_i − σ_j), and $ \underline{b}_{a} $ is a row vector of dimension 1 × (a − 1) of the form $ \underline{b}_{a} = \prod\nolimits_{j = 1}^{a} {B_{j} } $, j = 1, …, a, where, B _j matrices are non-square matrices such that $ B_{1} = 1,\,B_{2} = \sigma_{21}^{ - 1} ,\,B_{3} = \left[ {\begin{array}{*{20}c} {\sigma_{31}^{ - 1} } & { - \sigma_{32}^{ - 1} } \\ \end{array} } \right],\,B_{4} = \left[ {\begin{array}{*{20}c} {\sigma_{41}^{ - 1} } & 0 & {\sigma_{43}^{ - 1} } \\ 0 & {\sigma_{42}^{ - 1} } & {\sigma_{43}^{ - 1} } \\ \end{array} } \right],\,B_{5} = \left[ {\begin{array}{*{20}c} {\sigma_{51}^{ - 1} } & 0 & 0 & {\sigma_{54}^{ - 1} } \\ 0 & {\sigma_{52}^{ - 1} } & 0 & {\sigma_{54}^{ - 1} } \\ 0 & 0 & {\sigma_{53}^{ - 1} } & {\sigma_{54}^{ - 1} } \\ \end{array} } \right] $, and so on. The general formula for B _k (k ≥ 3) can be written as,

$$ B_{k} = \left[ {\begin{array}{*{20}c} {\sigma_{k1}^{ - 1} } \\ 0 \\ : \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\sigma_{k2}^{ - 1} } \\ : \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} : & 0 & 0 & {\sigma_{kk - 1}^{ - 1} } \\ : & 0 & 0 & {\sigma_{kk - 1}^{ - 1} } \\ : & : & : & : \\ : & {\sigma_{kk - 3}^{ - 1} } & 0 & {\sigma_{kk - 1}^{ - 1} } \\ : & 0 & {\sigma_{kk - 2}^{ - 1} } & {\sigma_{kk - 1}^{ - 1} } \\ \end{array} } \right]_{(k - 2) \times (k - 1)} $$

Note that due to the dimensions of the B _j matrices, the product $ \underline{b}_{a} = \prod\nolimits_{j = 1}^{a} {B_{j} } $ is a vector. We define the vector $ \underline{v}_{a} $ to be a column vector of the form,

$$ \underline{\nu }_{a} = \left[ {\begin{array}{*{20}c} {e^{{ - \sigma_{1} t}} - e^{{ - \sigma_{a} t}} } \\ {e^{{ - \sigma_{2} t}} - e^{{ - \sigma_{a} t}} } \\ {e^{{ - \sigma_{3} t}} - e^{{ - \sigma_{a} t}} } \\ : \\ {e^{{ - \sigma_{a - 1} t}} - e^{{ - \sigma_{a} t}} } \\ \end{array} } \right]_{(a - 1)x1} . $$

Appendix 3: Analysis of the case with constant mortality amongst stages

To see that the assumption of equal mortality at each stage cased the mortality rates in Eq. 5 to cancel out mathematically, consider the case where each μ_i is a constant μ in our solution Eq. 5. Then note that in Eq. 5, σ_ij becomes independent of μ, and as a result $ \underline{b}_{a} $ also becomes independent of μ. Further, in $ \underline{\nu }_{a} $, $ (e^{{ - \sigma_{i} t}} - e^{{ - \sigma_{a} t}} ) $ can be written as $ e^{ - \mu t} \left( {e^{{ - \gamma_{i} t}} - e^{{ - \gamma_{a} t}} } \right) $ for each element i. Thus, in the dot product $ (\underline{b}_{a} .\underline{\nu }_{a} ) $ in the Eq. 5, the term $ e^{ - \mu t} $ can be separated out as a multiplier, and after redefining, $ n_{a} (t) = e^{ - \mu t} \left( {\prod\nolimits_{i = 1}^{a - 1} {\gamma_{i} } } \right)(\underline{{\overline{b} }}_{a} .\underline{{\overline{\nu } }}_{a} ) $, such that term $ \left( {\prod\nolimits_{i = 1}^{a - 1} {\gamma_{i} } } \right)\left( {\underline{{\overline{b} }}_{a} .\underline{{\overline{\nu } }}_{a} } \right) $ becomes independent of μ. i.e. $ \underline{{\overline{b} }}_{a} = \underline{b}_{a} $ and $ \underline{{\overline{\nu } }}_{a} = \underline{\nu }_{a} $ for the special case where μ_i = 0 for all stages i. Now, we can write the proportion of each stage a that remains at time t, z _a(t), with respect to the total population at t:

$$ z_{a} (t) = n_{a} (t)/\sum\limits_{i = 1}^{s} {n_{i} (t)} = \left( {\prod\limits_{i = 1}^{a - 1} {\gamma_{i} } } \right)\left( {\underline{{\overline{b} }}_{a} .\underline{{\overline{\nu } }}_{a} } \right)/\sum\limits_{j = 1}^{s} {\left( {\prod\limits_{i = 1}^{j - 1} {\gamma_{i} } } \right)\left( {\underline{{\overline{b} }}_{j} .\underline{{\overline{\nu } }}_{j} } \right)} $$

where, s is number of stages. Thus, this equation is independent of μ. The numerator of this equation is n _a(t) for the case where μ_i = 0 for all stages for any t. The denominator is the solution to $ \sum\nolimits_{i = 1}^{s} {n_{i} (t)} $ for the special case where μ_i = 0 for all stages at any t if the population starts from 1 egg, thus remains 1 at any t. Hence, this can be simplified, so that,

$$ z_{a} (t) = \left( {\prod\limits_{i = 1}^{a - 1} {\gamma_{i} } } \right)\underline{{\overline{b} }}_{a} .\underline{{\overline{\nu } }}_{a} $$

(6)

which, is equivalent to z _a(t) = n _a(t) when μ_i = 0 for all stages at any t. Therefore, z _a(t) can be equated with the stage sizes normalized at each time step t in experimental data found in the literature which makes the assumption that μ_i = μ for all i = 1 to s.

Appendix 4: Fitting Eq. 2 to data using multiple substages

To derive solution to the modified system of equations in Eq. 1 by adding k sub-stages to each stage required using Laplace transformations. It yielded a complicated analytical result. Instead, we modified Eq. 2 to include sub-stages within stages, by assuming small differences in maturation rates among sub-stages. However, the solution in Eq. 2 cannot be simply transformed into a general case for the system to have multiple sub-stages, because in such case the denominator of the solution in Eq. 2 becomes zero, mathematically, as σ_ij = 0 when i and j were redefined for sub-stages in each stage, such that σ_i = σ_j. Therefore, we implemented the sub-stages for a given stage a by adding and subtracting a small constant (ɛ) to γ_a such that ɛ ≪ γ_a. For example, separating γ_a into three sub-stages would involve splitting γ_a among the three sub-stages, such that maturation rates were γ_a = > [γ_a − ɛ, γ_a, γ_a + ɛ]. Then we estimated γ_a using the modified Eq. 2 fitting to data from Uye et al. (1983) for small values of ɛ.

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Authors and Affiliations

Harshana Rajakaruna
- 1
Email author
Carly Strasser
- 2
- 3
Mark Lewis
- 1
- 2

1.Centre for Mathematical Biology, Department of Biological SciencesUniversity of AlbertaEdmontonCanada
2.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
3.Department of OceanographyDalhousie UniversityHalifaxCanada

Identifying non-invasible habitats for marine copepods using temperature-dependent R ₀

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Identifying non-invasible habitats for marine copepods using temperature-dependent R 0

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Identifying non-invasible habitats for marine copepods using temperature-dependent R ₀