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Model Structure:
An individual maximizes reproductive success at each time step.
Each time step in the model (t) corresponds, for simplicity, to a
single reproductive period. There are several state variables that
affect the individual’s reproductive success: the energy state of the
individual, xt; the circulating hormone signal, Ct; and the sensitivity
at each target tissue, 1 through i (S1,t,…, Si,t). An individual starts
with a certain state value for its energy state, X0, circulating
hormone level (C0) and sensitivity at each target tissue, 1 through i
(S1,0,…, Si,0). An individual’s state at each time step t is defined by
the state vector:
The organism must solve the problem of how to optimally
allocate its limited energy resources to maximize its fitness payoffs
(Wt). The trait value (Vi,t) at each tissue and the selection on the trait
(zi) determine the fitness payoffs to the organism.
𝑊𝑡 = 𝛽𝑡 (𝑉𝑔,𝑡 )
𝑧𝑔,𝑡
(𝑉𝑚,𝑡 )
𝑧𝑚,𝑡
(𝑉𝑝,𝑡 )
𝑧𝑝,𝑡
Several stochastic variables drive the simulation as well. The
mortality rate (µ) is a fixed chance that the individual will not survive
the given time step. Reproductive efficacy is modeled as a betadistributed random variable (β).
𝑋𝑡
𝐶𝑡
𝑥̅ 𝑡 = 𝑆1.
..
.
[ 𝑆𝑖 ]
The individual maximizes lifetime reproductive success by
choosing the optimal adjustment in hormone levels (ΔCt) and
sensitivities (ΔSi,t) at each time step. Constraints exist on the
available energy of the individual (i.e. it must be above a certain
level for any reproductive success (G, xrep), and it must be above
zero for the individual to survive).
An individual’s reproductive success in a given time step is a
function of multiple hormonally-mediated characters. The value (Vi,t)
of each trait is a function of the sensitivity of the target, Si,t and the
circulating hormone signal, Ct. The relationship between these
variables and the biological response is described by the simplified
version of the Michaelis-Menken equation, according to the
following expression:
𝑉𝑖,𝑡 =
𝑆𝑖,𝑡 𝐶𝑡
𝐾 + 𝐶𝑡
In the specific case being considered, we assume that
reproductive success is a function of three hormonally-mediated
characters: gamete maturation, mating effort and parental effort
(Vg,t, Vm,t, and Vp,t).
Parameter terms for the dynamic state model
State Variables
x
Energy state
Ct
Circulating level of hormone at time t
Si
Sensitivity at target tissue i
Static Parameters
G
Minimum hormonal state permissive of gamete maturation
xrep
Minimum body condition permissive of reproduction
|ΔCmax|
Absolute value of the maximum possible change of Ct
|ΔSi, max|
Absolute value of the maximum possible change of Si
zi
Selection index on trait i
γi,t
Cost of effort expended on each trait i
τ
Food availability, e.g. encounter rate
K
Michaelis-Menten constant, e.g. dissociation rate
η
Cost of hormone production
Stochastic Parameters
β
Random state variable reflecting reproductive efficacy at
time t
µ
Mortality probability