May 20, 2018, Sunday

# MBW:A New Model for Age-Size Structure of a Population

## Executive Summary

This paper derives a continuous population model that is dependent on age and mass (or some other behavior defining physiological characteristic). An analytic solution is found, and the model generated is then compared to other existing models.

## Sinko and Streifer's Model

### Definition of Parameters

Sinko and Streifer begin by defining $\eta (t,a,m)$ as a density function that depends on $t$ (time), $a$ (age), and $m$ (behavior defining physical attribute). The total number of animals between the ages $a_{0}$ and $a_{1}$ and the masses $m_{0}$ and $m_{1}$ are therefore equivalent to the following:

$\int _{{m_{{0}}}}^{{m_{{1}}}}\!\int _{{a_{{0}}}}^{{a_{{1}}}}\!\eta (t,a,m)\,{\mathrm {d}}a\,{\mathrm {d}}m$ [1]

Relaxing m0 and a0 to zero while also letting m1 and a1 go to infinity will result in the total number of animals in the population. The age and size based model is based on the following partial differential equation:

${\frac {\partial \eta (t,a,m)}{\partial t}}+{\frac {\partial \eta (t,a,m)}{\partial a}}+{\frac {\partial (g(t,a,m)\eta (t,a,m)}{\partial m}}=-D(t,a,m)\eta (t,a,m)$ [2]

Where $g(t,a,m)$ is the growth rate at time $t$ for an animal at age $a$ with mass $m$, and $D(t,a,m)$ is the death rate for specimens of age $a$ at time $t$ with mass $m$. Sinko and Streifer present a lengthy derivation showing that $\eta$ satisfies the above PDE.

In order for this problem to be well set (and therefore solvable), boundary conditions must be defined. Specifically,

$a(a,m)=\eta (0,a,m)$

$\mathrm{B} (t,m)=\eta (t,0,m)$[3]

Where $a(a,m)$ is the number of animals of age $a$ and mass $m$ that initially exist at time $t=0$, and $\mathrm{B} (t,m)$ is the number of newborn animals with mass $m$ that exist at time $t$. $\mathrm{B} (t,m)$ is often defined in the following manner:

$\mathrm{B} (t,m)=\int _{0}^{{\infty }}\!\int _{0}^{{\infty }}\!f(t,a,m',m)\eta (t,a,m'){\mathrm {d}}a{\mathrm {d}}m'$[4]

Where $f(t,a,m',m)$, ${\mathrm {d}}m$ is the rate at which animals of mass Failed to parse (lexing error): m’

and age $a$ give birth to babies of masses $m$ to $m+{\mathrm {d}}m$.


### Analytic Solution

An analytic solution to the PDE described above exists under special cases. If we assume that $g(t,a,m)$ is actually a function of $t$ and $a$ only, and that $D$ is independent of $\eta$, the following analytic solution can be obtained:

$\eta (t,a,m)=a[a-t,m]-\int _{0}^{t}\!g(t',t'+a-t){\mathrm {d}}t'\times$

$\exp {(-\int _{0}^{t}\!D[t',t'+a-t,m-\int _{{t'}}^{t}\!g(t'',t''+a-t){\mathrm {d}}t'']{\mathrm {d}}t')},a>t$

$\eta (t,a,m)=\beta [t-a,m-\int _{{t-a}}^{t}\!g(t',t'-t+a){\mathrm {d}}t']\times$

$\exp {(-\int _{{t-a}}^{t}\!D[t',t'-t+a,m-\int _{{t'}}^{t}\!g(t'',t''-t+a){\mathrm {d}}t'']{\mathrm {d}}t')},a[5]

This equation completely describes the ages and masses of all animals in the population provided that $a$, $\beta$, $g$, and $D$ are defined. Assuming that $g$ and $D$ are independent of $\eta$ does restrict this model. It essentially means that the population is not subject to “density effects.”[6] Food limitations and deaths due to increased encounters are density effects that would affect $g$ and $D$ respectively. As discussed earlier, $\beta$ is also dependent on $\eta$ if the population is generating offspring. If this is the case, the solution found for Sinko and Streifer's PDE is not complete.[7] In the region $a>t$, the solution is still known, but in the solution $a, $\beta$ depends on an integral of $\eta$, which makes the definition of $\eta$ implicit. This makes finding an analytic solution for $\eta$ difficult.[8]

## Relation to Other Models

### Oldfield's Model

D.G.Oldfield (1966) proposed an equation similar to the PDE model presented above. The key differences are that Oldfield’s three variables represent age and two physiological properties. His rate functions are also defined in a way that they only depend on time and the variable that they determine. Following Oldfield’s method will also generate an analytic solution to Sinko and Streifer's model given the following circumstances:[9]

1. $G$ is a function of $t$ and $m$ only
2. $D$ is not a function of $\eta$
3. Each animal has the same maximum lifespan, and at the end of the lifespan, the animal cleaves into two identical neonates (newborns).
4. A neonate has the same value of m as its parents.

The fourth assumption implies that $m$ cannot be mass. Sinko and Streifer note that other analytic solutions probably exist, but that they will most likely place serious restrictions on the population. Sinko and Streifer go on to say that numerical solutions are the only existing means for solving the original PDE given a realistic case.

### Von Foerster's Model

Sinko and Streifer's PDE includes H. Von Forester’s equation (1959) as a special case:

${\frac {\partial n}{\partial t}}+{\frac {\partial n}{\partial a}}=-D(t,a)n(t,a)$[10]

A key difference between Sinko and Streifer's model and Von Foerster's model is that Von Foerster's model is mass independent.

### Bailey's Parasite/Host Model

Von Foerster's model is the special case of V.A.Bailey's parasite/host model (1931) when there are zero parasites.[11] Sinko and Streifer provide a brief derivation in their paper.

### Leslie and Lewis's Model

P.H.Leslie (1945) and E.G.Lewis's (1942) population model is a discrete model that depends on time and age. Sinko and Streifer show that as the amount of time in between time steps goes to zero, Leslie and Lewis's model reduces to Von Foerster's model. [12]

### Frank's Model

P.W.Frank (1960) used P.H.Leslie’s model with $b$ and $p$ dependent on the total number of animals in an unsuccessful attempt to describe Daphnia pulex populations.[13] Frank then generalized the model by adding equations to find the mass of the individual, and therefore the biomass of the population. Animal mass is defined by:

$m(t+\Delta ,a+\Delta )=m(t,a)+g_{{\Delta }}[M(t),a]$

$m(t,0)=m_{0}$[14]

Where $g_{{\Delta }}$ is the increase in mass in the time $\Delta$ for an animal of age $a$ in a population with biomass $M(t)$. $M(t)$ is defined as:

$M(t)=\sum _{{i=1}}^{k}\!n(t,i*\Delta )m(t,i*\Delta )$[15]

Sinko and Streifer go on to show that as $\Delta t\to 0$, Frank's biomass model reduces to their own model. A key difference between the two models is that in Frank's model, any animal of a given age has the same mass, while Sinko and Streifer's model allows for a mass distribution among animals of the same age.[16]

### Slobodkin Model

L.B.Slobodkin (1935) has also proposed an algebra of population growth in which anmials are divided into different size and age categories. Animals in Slobodkin’s model either grow bigger, stay the same, or die. This is a significant deviation from Sinko and Streifer's model, since in their case animals are allowed to grow or shrink due to environmental factors.[17]

### Verhulst Model

A very simple logistic model presented by P.F.Verhulst (1838) and L.Pearl and L.J.Reed (1938) can be derived from the Von Foerster model as well. Verhulst's model reads as follows:

${\frac {{\mathrm {d}}N(t)}{{\mathrm {d}}t}}=r_{0}N(t)[1-{\frac {N(t)}{K}}]$[18]

Where $r_{0}=b-d$ is the intrinsic rate of increase, and $K$ is the equilibrium population size. $N(t)$ is the population size at time $t$. One can quickly infer from this model that population growth/decay is a function of density only. In the process of showing that Verhulst's model is a special case of Von Foerster's model, Sinko and Streifer also show that Verhulst's model assumes animals of all ages die at the same rate, and that birth rate is independent of animal density.[19]

## References

1. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
2. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
3. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
4. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
5. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
6. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
7. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
8. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
9. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
10. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
11. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
12. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
13. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
14. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
15. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
16. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
17. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
18. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918
19. J.Sinko and W.Streifer, Ecology, Vol. 48, No. 6 (Nov. 1967), pp. 910-918