Unification Theory of Optimal Life Histories and Linear Demographic Models in Internal Stochasticity

Life history of organisms is exposed to uncertainty generated by internal and external stochasticities. Internal stochasticity is generated by the randomness in each individual life history, such as randomness in food intake, genetic character and size growth rate, whereas external stochasticity is due to the environment. For instance, it is known that the external stochasticity tends to affect population growth rate negatively. It has been shown in a recent theoretical study using path-integral formulation in structured linear demographic models that internal stochasticity can affect population growth rate positively or negatively. However, internal stochasticity has not been the main subject of researches. Taking account of effect of internal stochasticity on the population growth rate, the fittest organism has the optimal control of life history affected by the stochasticity in the habitat. The study of this control is known as the optimal life schedule problems. In order to analyze the optimal control under internal stochasticity, we need to make use of “Stochastic Control Theory” in the optimal life schedule problem. There is, however, no such kind of theory unifying optimal life history and internal stochasticity. This study focuses on an extension of optimal life schedule problems to unify control theory of internal stochasticity into linear demographic models. First, we show the relationship between the general age-states linear demographic models and the stochastic control theory via several mathematical formulations, such as path–integral, integral equation, and transition matrix. Secondly, we apply our theory to a two-resource utilization model for two different breeding systems: semelparity and iteroparity. Finally, we show that the diversity of resources is important for species in a case. Our study shows that this unification theory can address risk hedges of life history in general age-states linear demographic models.


Feynman-Kac formula
Feynman-Kac formula is usefully to analyze dynamics of diffusion process in a potential medium. In a sense, the formula represents a differentiation formula which is extended to stochastic differential such that d (φ (X a ) S (a)) = dφ (X a ) S (a) + φ (X a ) dS (a) , (S. 1) where φ ∈ C ∞ (A) Then, it becomes from a property of stochastic differential. The Feynman-Kac formula asserted that the expectation of Eq.(S. 2) holds Additionally, this formula can extend to Dirichlet's boundary problem, such as F S (x), in the distribution sense. The proof of the formula is in [1,2].

(S.5)
Then, we obtain a Fokker-Planck equation with respect to the new population density,P ε (x → y), is given as follows: To derive path integral from Eq.(S.6), one uses Fourier transform of the functionP ε (x → y) with respect to y, such that When Eq.(S.6) is substituted into Eq.(S.6), we have On the RHS, integration by parts is applied, such that and expandḡ (ε, y),c j,j ′ (ε, y), andμ (ε, y) into a power series with respect to ε as follows: (S.10) Substituting (S.10) into (S.9), we obtain a transition rate for the sufficiently short time, ε, given by ∫ Substituting (S.11) into (S.8) and solving the ODE, we obtain the solution which iŝ Using inverse transform of the above equation,P ∆ε (x → y) becomes and from being Markovian process in the dynamics at finite time, we show where ∆ε = ε τ +1 − ε τ (ε τ +1 > ε τ > 0) and ε 0 = 0. It takes the limit of ∆ε to zero, then T ∆ε conserves a constant ε such that Accordingly, the limiting function expresses the summation over every projection function of stage transition which connect x with y at time ε, and that is the extended path-integral. We rewrite (S.15) asK whereẊ τ represents the differential ofX τ with respect to τ and where Taking t > a into account and changing the original coordinate, (t, a), to the new coordinate, (t − a, 0) in Eq.(S.16), we obtain Setting ε = a, we consequently have the path-integral expression of Eq.(S.4) as follows: We then set lim , · · · ,g d (a, X a ) ) be a vector to simplify notations and to derive another expression of the projection function. The Hamiltonian, then, can be written as a quadratic form, such that H Note that terms of the RHS including q in Eq.(S.13) is identical to a characteristic function of multidimensional normal distribution, hence those terms represent the Fourier transform of the distribution. Consequently, another expression (the Lagrangian expression) is composed of the inverse transform of Eq.(S.13) and which is where Z ′ τ denotes normalization constant. This expression is used in the analysis of classical state transition curve (no stochasticity) [3].
The RHS of Eq.(S.21) describes the sum over all possible transitions of life history starting from initial state P t−a (0, y 0 ) to P t (a, y m ). In other word, one expects similar relationship between Eq.(19) and Eq.(S.21) in the concept of path-integral as well. Since each vital rate of K generally includes survival rate (p a,m ), we decompose the vital rate k a,m ′ m to a pure transition probability k ′ a,m ′ m and the survival rate as follows: Substituting this into Eq.(S.20b) and assuming Y a to be a stochastic process of size growth generated by the following parameterized collection of random variables Eq. (27), we can rewrite the equation as follows: where components are When P * (0, y 0 ) is the first state in eigenvector of L, the above equation is rewritten by using the eigenvalue, λ 0 , as follows: Since P * (0, y 0 ) of both sides can be canceled, we have the characteristic equation of L This characteristic equation obviously parallels Eq.(9), and is nothing but the generalized Euler-Lotka equation in TMM to age-size model. Furthermore, k a,m ′ ,m is assumed to follow Gaussian distribution associated with mortality at an age interval ∆a, such that Rewriting Eq.(S.27) for the characteristic function form, we havē (S.28) We can adopt those transition rates to the projection function as follows: Our model is on the hypothesis that difference of vital rates yield to the Gaussian distribution at infinitesimal short age interval ∆a. Consequently, path-integral model is an expression of LDMs in the sense that one of expression has correspondence with those of the others if the temporal development of life histories can be approximated by local Gaussian process (Ito process), such as Eq.(1).

S3
Viscosity solution is an important idea to consider Hamiltonian systems and is introduced in 80's [4]. The idea unifies Hamiltonian in population vector and in control theory.

Definition of viscosity solutions
We set the function where S d denotes a set of all real symetric matrices. When H is degenerate elliptic in E, it satisfies where a ∈ [0, α), x ∈ A, p ∈ R d , Q ∈ S d , and w ∈ R + . If the function is a monotonic function in w, it satisfies H (a, x, p, Q, w 2 ) .

For example, Hamiltonian
is degenerate elliptic, monotonically increasing function. This function is the same Hamiltonian used in the stochastic maximum principle [5]. Then, p, Q, and w represent co-state variables in the principle. Let H be degenerate elliptic and monotonic function. A sub-solution in the viscosity sense, is defined by ψ −ψ (ψ ∈ C 2 (A)) having a maximum value of zero atx andψ satisfying We then use abbreviation of derivatives: .
A super-solution of Eq.(S3.2) in the viscosity sense, ψ ∈ C (A), is defined by ψ −ψ (ψ ∈ C 2 (A)). It has a minimal value of zero atx andψ satisfying When ψ satisfies the sub-and super-solution in the viscosity sense, it is referred to as a "viscosity solution" [6]. Additionally, the viscosity solution of the nonlinear evolutional PDE can be defined as the sub-solution of When w satisfies the sub-solution and the super-solution of the above PDE in the viscosity sense, it is called a "viscosity solution" of the PDE. Extension of the solutions by viscosity solutions unifies two approaches of analysis involved in the HJB equation and the maximum principle in control theory [7]. One can then find the correspondence of (p, Q, w) to and shows that this Hamiltonian is the conjugate of the Hamiltonian in the path integral, Eq.(19), with respect to −iq τ . Consequently, the fittest has a minimum Hamiltonian in its habitat, and the Hamiltonian naturally appears in our formulation.

S4
Derivation of a general ψ v λ (x) in semelparous species Since X a has strong Markov property that is a property of Ito's SDE [1], we can show the following relationship: . ↑ using strong Markov property

Mature age density of semelparous species
Since the optimal utilization is constant, we can use an adjoint Hamiltonian generated by geometric Brownian motion as follows:H This means, therefore, that statistics of Eq.(59) come down to those of Brownian motion via the above heat equation. The mature age density then is converted to the first passage time problem of the Brownian motion. When converted mature size is z * := log x * , the probabilty of B a > z * , P z (B a > z * ), satisfies the following equation by using the fundamental solution in Eq.(S5.4) If a * z * denotes the first passage time of z * , it is known as "reflection principle" [2] that Therefore, that is Consequently, the semelparous mature age density that all parameters appear follows A S (a) = = log x *