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Diel vertical migration arising in a habitat selection game

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Abstract

Predator and prey react to each other, adjusting their behavior to maximize their fitness and optimizing their food intake while keeping their predation risk as low as possible. In a pelagic environment, prey reduce their predation mortality by adopting a diel vertical migration (DVM) strategy, avoiding their predator during their peak performance by finding refuge in deep layers during daylight hours and feeding at the surface during the night. Due to the duality of the interaction between prey and predator, we used a game theory approach to investigate whether DVM can be a suitable strategy for the predator as well as the prey. We formulated three scenarios in plankton ecology in order to address this question. A novel finding is that mixed strategies emerge as optimal over a range of the parameter space, where part of the predator or prey population adopts a DVM while the rest adopt one or other “sit and wait” strategies.

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Acknowledgements

This work was supported by the Greenland Climate Research Centre and the VKR Centre for Ocean Life.

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Authors and Affiliations

  1. National Institute of Aquatic Resources, Technical University of Denmark, Kavalergaarden 6, 2920, Charlottenlund, Denmark

    Julie Sainmont, Uffe H. Thygesen & André W. Visser

Authors
  1. Julie Sainmont
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  2. Uffe H. Thygesen
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  3. André W. Visser
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Corresponding author

Correspondence to Julie Sainmont.

Appendix

Appendix

Solution scheme

The Nash equilibrium of the game can be found algebraically, by requiring that all strategies which are adopted by a positive fraction of animals share the same fitness, and that all strategies which are not adopted, have no greater fitness. This leads to a set of linear equations. However, this approach is somewhat tedious, because one must treat the boundaries (i.e., solutions where some strategies are not adopted) separately. A more convenient and flexible approach is to use that the Nash equilibrium is necessarily an equilibrium of the replicator equation (see Hofbauer and Sigmund 2003, for background and a precise converse statement).

With this approach, the replicator equation governs the dynamics of the fractions of the different strategies as follows: The fitness of prey (Eq. 2) and of predator (Eq. 3) are used as growth rates of the subpopulations which adopt each strategy. These dynamics do not necessarily mimic real population dynamics, but is merely a computational method to identify the Nash equilibrium, by marching the replicator equation forward in time until steady state. We formulate the replicator equation in discrete time. In a first step, populations grow according to their fitness:

$$ \begin{array}{rll} &&{\kern-9pt} \begin{cases} N_S'(i+1)=N_S(i)+N_S(i)F^+_{N_S}dt \\ N_D'(i+1)=N_D(i)+N_D(i)F^+_{N_D}dt \\ N_m'(i+1)=N_m(i)+N_m(i)F^+_{N_m}dt \end{cases}\notag\\ &&{\kern-9pt}\begin{cases} P_S'(i+1)=P_S(i)+P_S(i)F^+_{P_S}dt \\ P_D'(i+1)=P_D(i)+P_D(i)F^+_{P_D}dt \\ P_m'(i+1)=P_m(i)+P_m(i)F^+_{P_m}dt \end{cases} \end{array} $$
(8)

In the next step, the abundance proportions are renormalized so as to sum to one:

$$ \begin{array}{rll} &&{\kern-9pt}\begin{cases} N_S(i+1)=\dfrac{N_S'(i+1)}{N_S'(i+1)+N_D'(i+1)+N_m'(i+1)} \\ N_D(i+1)=\dfrac{N_D'(i+1)}{N_S'(i+1)+N_D'(i+1)+N_m'(i+1)} \\ N_m(i+1)=\dfrac{N_m'(i+1)}{N_S'(i+1)+N_D'(i+1)+N_m'(i+1)} \end{cases}\\ &&{\kern-9pt}\begin{cases} P_S(i+1)=\dfrac{P_S'(i+1)}{P_S'(i+1)+P_D'(i+1)+P_m'(i+1)} \\ P_D(i+1)=\dfrac{P_D'(i+1)}{P_S'(i+1)+P_D'(i+1)+P_m'(i+1)}\\ P_m(i+1)=\dfrac{P_m'(i+1)}{P_S'(i+1)+P_D'(i+1)+P_m'(i+1)} \end{cases} \end{array} $$
(9)

This completes the recursion, which is then iterated until steady state.

Stabilization

The Nash equilibrium is an equilibrium of the replicator dynamics, but not necessarily an asymptotically stable equilibrium. Since our model of fitness does not include a direct dependence of the density of conspecifics, the replicator dynamics may display periodic dynamics which cycle around the Nash equilibrium, similar to the classic Lotka–Volterra system. To stabilize the Nash equilibrium and dampen out these cycles, we modify the replicator equation as follows: We add a proportion “a” of the difference between the last two time steps of the predators proportion in the surface (P S (i − 1) − P S (i − 2)), to the proportion of prey in the surface (N S ):

$$ N_S'(i+1)=N_S(i)+N_S(i)F^+_{N_S}dt + a(P_S(i)-P_S(i-1)) $$
(10)

This computational stabilization mimics damping in physical systems and does not change the system equilibrium value, as at equilibrium, the predator proportion does not change anymore (P S (i) = P S (i − 1), so P S (i) − P S (i − 1) = 0). Again, we stress that this is merely a computational method for identifying the Nash equilibrium, so an ecological interpretation of this damping term is not necessary.

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Sainmont, J., Thygesen, U.H. & Visser, A.W. Diel vertical migration arising in a habitat selection game. Theor Ecol 6, 241–251 (2013). https://doi.org/10.1007/s12080-012-0174-0

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