Spatial Velocity

The spatial transport velocity of fish from species κ is given by:

q[κ]=

  

Vcurrents if r ă C[κ]_C/dmw[κ]_forage Vcurrents+V_reactive[κ] +V_predictive[κ] otherwise

(11.46)

For simplicity, we assume that fish which do not forage on their own have no active movement and are transported in space solely by currents (Vcurrents). All other individuals are also affected by Vcurrents but exhibit two additional movement terms that together represent their active locomotion.

In modeling the active movement of individuals, we follow the approach of Fernö et al. (1998), who describe fish swimming as being governed by two processes (see also Neill 1979). The first process is reactive movement, which means that the fish choose a swimming speed and direction based on their immediate surrounding. Thereby, they react to approaching predators, from which they flee, or to prey patches, by which they are attracted. The second process is predictive movement, which is based not on stimuli from the current environment of the fish but rather on its experience and its instinct (regarding learning in fish, see Brown et al. 2008). Examples of this second type of movement are feeding and spawning migrations.

Currents

If a body with zero velocity is put into a flow field with constant velocity everywhere, the difference in speed between the flow medium and the body will decay exponentially in time (Rieutord 2014). Therefore, we set Vcurrents to be identical with the velocity field supplied by the ocean model with which the Sprat Model is coupled.

11.2. The Fish Model

Reactive Movement

Reactive movement can be modeled based on a habitat index function H[κ], which describes how favorable a location is to an individual fish (cf. Bertignac et al. 1998). Such a habitat index function can consider the amount of predators and prey at any given place as well as other environmental parameters, such as temperature and dissolved oxygen concentration. Using the index function, the fish can be modeled as trying to locally maximize H[κ]by swimming in a certain direction.

A well-known functional form for H[κ]was introduced by Gilliam and Fraser (1987), who model fish as seeking out locations with the highest ratio of foraging rate to mortality rate, which implies

H[κ]= c

[κ] prey

c[κ]_pred. (11.47)

However, Railsback et al. (1999) point out several practical problems with this approach. First, the habitat index is undefined in the absence of preda- tors. Second, it underestimates the importance for the fish to avoid predation in comparison to the importance of foraging as both are equally relevant in Equation 11.47. Instead, fish would likely accept a period of starvation in order to avoid being killed. These concerns can be addressed by intro- ducing an additive constant into the denominator of Equation 11.47 and a scaling exponent that influences the importance of predator avoidance in comparison to foraging: H[κ]= c [κ] prey (c[κ]_pred+C)α (11.48)

An alternative to the functional forms of Equations 11.47 and 11.48 is to ignore the risk of predation and simply choose

H[κ]=cprey[κ] . (11.49)

Since this approach of ignoring predation risk has been successfully em- ployed by Radtke et al. (2013) in their end-to-end model for the Baltic Sea,

we opt for defining the habitat index function for our model as given in Equation 11.49.

In order to let the fish find the habitat that is locally optimal for them, it has been proposed (e. g., by Bertignac et al. 1998) to set reactive movement velocity proportional to the (spatial) gradient of the habitat index:

V[κ]_reactive=ξ0  B BxH [κ]_, B ByH [κ] _(11.50)

We found that this optimization strategy is unsuitable for our model because it is too locally confined and produces large accumulations of fish in local maxima of H[κ] that are relatively inadequate compared to other close-by locations. Therefore, we choose to assume that fish have perfect information about their surroundings within a certain radius rview and swim to an absolute maximum of H[κ]within this area.

The area of perfect information is given by

K(x, y) =n(v, w)PR2:k(v ´ x, w ´ y)kďrview o

XΩ_S. (11.51) Within its bounds, we are looking for the spatial point with maximum habitat index and choose our swimming direction as

δo[κ](t, x, y, r) = 

arg max_(v,w)PK(x,y)H[κ](t, v, w, r)´(x, y). (11.52) Equation 11.52 assumes that there is exactly one absolute maximum—if there is more than one, we select the one that minimizeskδo[κ]k. The problem of finding an absolute maximum in K(x, y)is computationally feasible since all densities are discretized for approximating the solution to Equation 11.37 (hence, we need to check only a finite amount of points to find the desired maximum).

Finally, we can define the reactive movement velocity as proportional to δo[κ]as

11.2. The Fish Model with ξ[κ]=    ς[κ]l[κ](r) kδ[κ] o k ifkδ[κ]o ką0 0 otherwise, (11.54)

where ς[κ] is the cruise speed of fish from species κ in body length per second.

As an alternative to supplying a constant cruise speed in body length per second, one could use the approach of Videler (1993), who models swimming speed using stride length and tail beat frequency at cruise speed. His approach has the advantage that tail beat frequency explicitly depends on environment temperature and body size. However, an important drawback (and the reason why we choose not to use it) is that there is little data published on stride lengths and tail beat frequencies at cruise speed for different fish species.

Predictive Movement

The fish in our model employ predictive movement strategies in two cases: 1. If prey abundance is too low within the radius of perfect information

rviewto sustain the fish, they initiate a feeding migration until they find more suitable habitat.

2. During the mating season, fish travel to spawning grounds.

Therefore, we define a migration velocity which is composed of feeding and spawning migration:

V_migration[κ] =V_feeding[κ] +V[κ]_spawning (11.55) If there is no reactive movement (because no location within rviewhas a higher habitat index than the current one) and there are more fish at the current location than can be sustained by the respective prey concentration, the fish have to migrate. This can be formalized as

V_feeding[κ] =

(

(1, 0) ifkV_reactive[κ] k =0 ^ m[κ]ą ζ0cprey[κ]

where ζ0is a linear scaling constant. The movement strategy we employ for feeding migrations is simple: swim in the direction of the positive x axis (the swimming speed will be adjusted later on). A similar strategy (“swim westwards”) has actually been observed in herring in the Norwegian Sea (Fernö et al. 1998). Of course, more sophisticated strategies can be used, such as swimming to known feeding locations etc.

For spawning migrations, we assume that there is a single spawning ground for each species given by its center s[κ] = (xs[κ], y[κ]s ). Furthermore, a fixed mating season is described by the subset S[κ] Ă [0, 1], where the interval represents time of year (with 0 being the first moment and 1 the last moment of the year). We introduce the function θ(t)P[0, 1]to convert simulation time t to time of year. With these prerequisites, the spawning migration velocity is given by

V[κ]_spawning=

(

(x[κ]s ´x, y[κ]s ´y) if θ(t)PS[κ]

0 otherwise. (11.57)

V_spawning[κ] formalizes that fish are attracted by the spawning region during the mating season. Its magnitude is larger the further away the fish are from the spawning ground (the actual swimming speed will be adjusted later on). The approach can, of course, be extended to multiple spawning areas represented by regions rather than by points if necessary.

If there is a reason for the fish to migrate (i. e.,kV_migration[κ] k ą 0), we use the predictive movement velocity to combine migratory and reactive movement: V[κ]_predictive=                ´V[κ]_reactive+ ς[κ]l[κ](r) V_reactive[κ] +V[_migrationκ] kV[κ] reactive+V [κ] migrationk ifkV_migration[κ] ką0 ^ kV_reactive[κ] +V[κ]_migrationką0 0 otherwise (11.58)

11.2. The Fish Model

11.2.5

Sources and Sinks

The source term of the model is given by

H[κ]=       

´m[κ]H_fishing[κ] +H_backgr[κ] +Hrepr[κ]

+E[κ]_I[κ]_´L[κ]_´R[κ] _{if r ě C}[κ] C/dmw

[κ] forage ´m[κ]H_backgr[κ] +Hrepr[κ] ´L[κ] otherwise.

(11.59)

For fish which are able to actively forage, the source term consists of losses due to fishing (H[κ]_fishing), background mortality (H[κ]_backgr), predation (L[κ]), and respiratory costs R[κ]. The intake of food from predation is described by the term E[κ]I[κ], where I[κ]is the gross carbon mass intake and E[κ] is the assimilation efficiency. The redistribution of mass during reproduction (from mature individuals to eggs) is covered by H[κ]repr.

For very early life stages that cannot forage on their own, only back- ground mortality, reproduction, and losses due to predation have to be considered.

Fishing

Extraction of individuals due to fishing is described by an instantaneous fishing mortality rate (cf. Chapter 3) which can vary depending on spatial location and fish size. For example, in our evaluation of the Sprat Model in Chapters 15 and 16, H_fishing[κ] is given by

H_fishing[κ] (t, x, y, r) =

(

F[κ](t) if M[κ]ě 1₂M[κ]_mature

0 otherwise, (11.60)

where F[κ](t)is the observed fishing mortality for species κ.

Background Mortality

To account for losses of fish due to effects that are not explicitly considered in our model (such as predation by birds and marine mammals), we introduce

the background mortality

H[κ]_backgr=

(

ε[κ]_B m[κ] if M[κ]ą1₄M[κ]mature

ε[κ]_B m[κ]´ ζ[κ]_B min(0,∆T(t)) otherwise. (11.61) For all individual sizes, we apply a quadratic instantaneous death term with mortality rate ε[κ]_D. Since earlier life stages are especially vulnerable to fluctuations in temperature, we include an additional linear death term for smaller individuals that depends on the deviation∆T(t)from the average habitat temperature (see Houde 2009).

Metabolic Costs

Respiratory costs are given by

R[κ]= m

[κ] r



R[κ]_S +R[κ]_A =u[κ]R[κ]_S +R[κ]_A , (11.62)

where R[κ]_S is the Standard Metabolic Rate (SMR) of an individual fish and R[κ]_A represents the additional respiratory costs due to swimming (net swimming costs). R[κ]_A corresponds to the Active Metabolic Rate (AMR) minus theSMR. According to Clarke and Johnston (1999),SMRis fitted well for many teleostei by

SMR= 1

5.43M

0.8 _(11.63)

with M being the wet mass of an individual in g and SMR in mmol O2h´1. In order to convert oxygen consumption to carbon losses, we make use of the so-called respiratory quotient RQ, which describes the ratio of exhaled CO2volume to inhaled O2volume:

RQ= 1 mol CO2

1 mol O2 (11.64)

While the value of RQ depends on the diet, Videler (1993) determines

11.2. The Fish Model

to be a good average value for fish. Therefore, it holds that

1 mol O2= 1 0.96 mol CO2, (11.66) which implies: 1 mmol O2h´1= 1 0.96 mmol CO2h ´1 _(11.67) = 1 0.96 mmol C h ´1 _(11.68) = 12 0.96 mg C h ´1 _(11.69) = 12 0.96 ¨ 106_¨602 kg C s ´1 _(11.70)

By converting to appropriate units and compensating for temperature changes using a Q10 temperature coefficient, we define theSMRof an indi- vidual fish in our model as

R[κ]_S =QSMR₁₀  ∆T(t) 10 12 5.43 ¨ 0.96 ¨ 106_¨602  103M[κ]0.8. (11.71) Videler (1993) deems QSMR

10 =2 to be appropriate in the context of resting metabolic rates of fish.

Regarding the respiratory net costs of swimming, Boisclair and Tang (1993) as well as Ohlberger et al. (2005) propose a model of the form

AMR=SMR+aMbvc, (11.72)

where M is wet mass in g, v is swimming speed in cm s´1, and AMR is in mg O2h´1. A parametrization of Equation 11.72 for steady swimming in many fish is a =10´2.43, b=0.8, and c= 1.21 (Boisclair and Tang 1993). For determining v, we only have to consider the velocity due to active movement

With regard to unit conversion, we use Equation 11.66 again to derive: 1 mg O₂h´1=1 mg O₂h´1¨ 1 32 mol O2 g O₂  (11.74) = 1 32 mmol O2h ´1 _(11.75) = 1 0.96 ¨ 32 mmol CO2h ´1 _(11.76) = 12 0.96 ¨ 32 ¨ 106_¨602 kg C s ´1 _(11.77)

Applying appropriate unit conversions, we can define the net swimming costs per individual as

R[κ]_A = 12 ¨ 10 ´2.43 0.96 ¨ 32 ¨ 106_¨602  103M[κ]0.8102kV[κ]_activek1.21. (11.78) Reproduction

During the spawning season S[κ]Ă[0, 1], mass is transferred from mature female fish to (fertilized) eggs. We assume a constant 50 % : 50 % sex ratio of females to males in the population. The net wet mass fecundity Φ[κ]_{describes how many fertilized eggs per kg wet mass a mature female} produces during the spawning season.

In order to describe the mass transfer process due to reproduction, we define the carbon mass at maturity

r[κ]_mature=c[κ]_C/dmc_d/wm[κ] M[κ]_mature, (11.79) the carbon mass of an egg

r[κ]egg=cC/dm[κ] w [κ]

egg, (11.80)

and the duration of the spawning season of species κ in seconds

θ[κ]s =365 ¨ 24 ¨ 602¨

Z 1

11.2. The Fish Model

where1_S[κ] is the characteristic function on S[κ]. This allows us to define the

biomass spawning rate

Bs[κ]=      1 2 Φ[κ]_r[κ] egg c[_C/dmκ] c[_d/wmκ] θs[κ] m[κ] if θ(t)PS[κ]^r ě r[κ]mature 0 otherwise, (11.82)

which describes the amount of egg biomass that is spawned at each moment of the spawning season (spawning happens at a constant rate during the whole season).

In order to reintroduce the egg biomass into the distribution m at an appropriate “place” in the size dimension, we define an insertion distribu- tion ψ[κ](r)that only has a small support which is centered around r[κ]egg. We require ψ[κ]:[rmin, rmax]ÑRě0to be continuous and to fulfill

arg max_{φP[rmin,rmax]}ψ[κ](φ)  =r[κ]egg (11.83) ^ Z r_max rmin ψ[κ](φ)dφ=1 (11.84)

^ ψ[κ](r) =0 for all r ě rmature[κ] . (11.85) The insertion distribution ψ[κ](r)allows us to describe the redistribution of egg mass as H[κ]repr(t, x, y, r) =ψ[κ](r) Z r_max r[_matureκ] B [κ] s (t, x, y, φ)dφ ´ B[κ]s (t, x, y, r). (11.86)

In the following lemma, we show that Hrepr[κ] does not interfere with the mass conservation property of the Sprat Model by introducing spurious biomass sources or sinks.

Lemma 11.1. The term Hrepr[κ] , as defined in Equation 11.86, conserves biomass as

In document Model-Driven Software Engineering for Computational Science Applied to a Marine Ecosystem Model (Page 190-199)