The migration game in habitat network: the case of tuna

Abstract

Long-distance migration is a widespread process evolved independently in several animal groups in terrestrial and marine ecosystems. Many factors contribute to the migration process and of primary importance are intra-specific competition and seasonality in the resource distribution. Adaptive migration in direction of increasing fitness should lead to the ideal free distribution (IFD) which is the evolutionary stable strategy of the habitat selection game. We introduce a migration game which focuses on migrating dynamics leading to the IFD for age-structured populations and in time varying habitats, where dispersal is costly. The model predicts migration dynamics between these habitats and the corresponding population distribution. When applied to Atlantic bluefin tunas, it predicts their migration routes and their seasonal distribution. The largest biomass is located in the spawning areas which have also the largest diversity in the age-structure. Distant feeding areas are occupied on a seasonal base and often by larger individuals, in agreement with empirical observations. Moreover, we show that only a selected number of migratory routes emerge as those effectively used by tunas.

Appendix: Model calibration for Atlantic bluefin tuna

Migration costs

The time needed to migrate between two habitats regulates the cost of migration in fish population since the energy consumed will be higher the longer is the migration time. The power rate consumed while swimming at an optimal speed (P) is:

$$\begin{array}{@{}rcl@{}} P=\alpha_{0} \mathit{w}^{\eta} U^{\beta} \end{array} $$

(9)

where w is the mass of the fish, while the estimates for allometric constants α ₀ and η assume fish swimming in a turbulent regime (i.e., high Reynolds number) (Ware 1978), Table 4).

Table 4 Scaling of physiological rates with size and parameter values for tuna from (1) (Overholtz 2006), (2) (Ware 1978), (3) (Dewar and Graham 1994), and (4) (Block and Stevens 2001)

We can assume that during migration fish swim at the optimal speed (U ^∗) at which the total energy expenditure per unit distance is minimized. Using an allometric function for the metabolic costs M = α ₁ w ^γ, a general form of U ^∗ can be derived by an optimisation procedure relating the swimming cost to the total cost of moving (metabolic cost plus power output):

$$\begin{array}{@{}rcl@{}} U^{*}=\left[\frac{-\alpha_{1} \mathit{w}^{\gamma}}{\alpha_{0} \mathit{w}^{\eta} (1 - \beta)}\right]^{\frac{1}{\beta}} \end{array} $$

(10)

where α ₁ and γ are allometric constants for fish metabolism (Table 4). This results in an allometric scaling for the optimal swimming speed as:

$$\begin{array}{@{}rcl@{}} U^{*} \approx \mathit{w}^{\frac{\gamma-\eta}{\beta}} \end{array} $$

(11)

In tuna, the exponent β has been found to range between 1.4<β < 2.8 (Dewar and Graham 1994) and we assume β = 2.1, which provides swimming speeds in the range reported for several tuna species (1.2 − 2.4 body length per second) (Block and Stevens 2001). Thus we obtain a scaling U ^∗≈w ^0.06.

Demography

Uncertainties exist on the definition of demographic parameters for the bluefin tuna population (Simon et al. 2012). In our model, the young-of-the-year stage (0–1 years) excludes the egg phase and does not have reproductive potential while at juvenile stage (1–5 years), a small fraction is mature for reproduction. The reproductive maturity increases up to 50 % at the adult stage (5–10 years) while mature (10–20) and old (20–35) stages are fully reproductive but the latter has a lower survival rate. Those rates are consistent with observed maturity at age data for western and eastern Atlantic bluefin tuna (SCRS 2012) and are used to define the values of r _k . Moreover, the value survival (q) and growth (g) values used in the Leslie matrix are consistent with reported values for the yearly mortality rates (SCRS 2012) and provide a realistic bluefin tuna age-structure (Fig. 7) with a maximum population growth rate (0.15) that is in the range of previous estimates (Simon et al. 2012). Finally, we constrain the global bluefin tuna population using a given total carrying capacity K _t and assume a density dependent function on the spawning factor s.

Fig. 7

Age structure data from the ICCAT assessment group (black) on bluefin tuna and from the model using the Leslie matrix estimates (red)

Extended sensitivity analyses

At low spawning intensity and high migration costs (Fig. 8g), only the spawning areas are occupied. Decreasing habitat selection costs allows tuna to migrate in adjacent feeding areas (G and C) but reduce the total biomass and increase fluctuations in the migration behaviour (Fig. 8a, d). On the other hand, at high spawning and low migration costs (Fig. 8a, b), the biomass reaches the total carrying capacity over few months, and all habitats are occupied although at different levels of biomass.

Fig. 8

Sensitivity of the population structure and total biomass in different habitats under different spawning intensity s and habitat selection cost μ in aseasonal environments