4. Exponential Decay - Mathematical Modeling for Life Sciences, A Primer

Abstract

This chapter develops both deterministic and stochastic models to analyze how population size changes over time under the assumption of an age-independent death rate in closed populations. Beginning with a distribution function describing population age structure, a convection equation governs aging in the absence of deaths. Introducing constant mortality leads to exponential decay of total size. A stochastic approach models population size as a random variable and relates transition probabilities, recovering the deterministic solution when averaging expected size. Solving for individual lifetime distributions links dynamics across scales through averaging. Extensions to age-dependent but stationary mortality demonstrate self-organized exponential decay, connecting perspectives from random interactions to emergent collective behaviors. The established foundations connect stochastic and deterministic views on simple populations and lay the groundwork for further complexity.

Introduction¶

Population dynamics is the study of how populations change over time and space due to various biological and environmental factors. Central topics in population dynamics include birth, death, immigration and emigration, density dependence, and biotic and abiotic interactions between individuals within a population or between species.

In this chapter, we will begin exploring population dynamics by studying a simple death process in closed populations. Specifically, we will develop both deterministic and stochastic models to analyze how population size changes over time under the assumption of an age-independent death rate. This forms the basis for understanding more complex population dynamics incorporating additional realistic features like age-structure, density dependence, and other stochastic birth and death mechanisms.

By establishing these foundations, we aim to connect individual-level stochasticity to emergent deterministic behaviors at the population scale.

Dynamics of Aging¶

Let us analyze the dynamics of a closed population with no births or deaths over time. We define the function $\eta(\tau, t) d\tau$ as the fraction of individuals in the population with ages in the interval $[\tau, \tau+d\tau]$ at time $t$ . This function fully describes the population state at any given time and how it evolves. Specifically, the total population at any time $t$ is given by:

N(t) = \int_0^\infty \eta(\tau', t) d\tau'.

(1)

In the absence of births, deaths, and migration, individuals can only age over time. This means the plot of the function $\eta(\tau, t)$ shifts to the right as time passes. We can mathematically express this as:

\eta(\tau, t+\Delta t) = \eta(\tau-\Delta t, t).

(2)

Expanding the left-hand side (LHS) and right-hand side (RHS) terms in a Taylor series about $t$ and $\tau$ , respectively, gives:

\eta(\tau, t) + \frac{\partial \eta}{\partial t} \Delta t + \dots = \eta(\tau, t) - \frac{\partial \eta}{\partial \tau} \Delta t + \dots.

(3)

Simplifying and taking the limit as $\Delta t \to 0$ yields the governing equation:

\frac{\partial \eta}{\partial t} = -\frac{\partial \eta}{\partial \tau}.

(4)

The RHS can be viewed as an age-related flux term. Therefore, this equation expresses conservation of population as the rate of change within an infinitesimal age interval equals the net outflux due to aging.

Since the total population $N(t)$ was assumed to remain constant over time due to no births, deaths, or migration, it follows that:

0 = \frac{d}{d t}\int_0^\infty \eta d\tau =\int_0^\infty \frac{\partial \eta}{\partial t} d\tau = - \int_0^\infty \frac{\partial \eta}{\partial \tau'} d\tau'.

(5)

The last equality expresses the fact that aging redistributes individuals among age classes but does not change the total population size.

Age-Independent Death Rate¶

We can modify Eq. (4) to account for deaths by including an additional death rate term

\frac{\partial \eta}{\partial t} = -\frac{\partial \eta}{\partial \tau} - \mu(\tau) \eta,

(6)

where $\mu(\tau)$ is the death-rate constant for individuals of age $\tau$ . It represents the probability per unit time that an individual of age $\tau$ dies.

Integrating Eq. (6) gives:

\frac{dN}{dt} = - \int_0^\infty \mu(\tau') \eta(\tau', t) d\tau'.

(7)

Now assume that the death rate is independent of age. Then:

\frac{dN}{dt} = - \mu N.

(8)

This differential equation has the well-known solution of exponential decay:

N(t) = N_0 e^{-\mu t},

(9)

where $N_0$ is the initial population size. This demonstrates that under the assumption of a constant, age-independent death rate, the population will decrease exponentially over time.

An important parameter that arises from this solution is the population half-life ( $t_{1/2}$ ), defined as the time required for the population to decrease to half its initial value. Specifically:

\frac{N_0}{2} = N_0 e^{-\mu t_{1/2}}.

(10)

Solving this equation for $t_{1/2}$ gives:

t_{1/2} = \frac{\ln{2}}{\mu}.

(11)

Stochastic Description¶

In the previous section, we modeled population dynamics deterministically and showed the population decays exponentially over time when death rates are independent of age. Here, we develop a stochastic framework to analyze this process more rigorously.

Let $n$ be a random variable representing population size. Define $P(n=N;t)$ as the probability that the population has a value of $N$ individuals at time $t$ :

P(n=N;t).

(12)

Additionally, let $\mu$ denote the probability per unit time that an individual dies independently of others. Considering these variables, the evolution of $P(n=N;t)$ is described by:

P(n=N;t+\Delta t) = P(n=N;t)(1-\mu N\Delta t) + P(n=N+1;t)\mu(N+1)\Delta t.

(13)

This equation accounts for two possibilities that could result in a population of size $N$ over a short time interval $\Delta t$ . 1) the population was already size $N$ at time $t$ and no deaths occurred. 2) The population was size $N+1$ at time $t$ and one death occurred. We assume that $\Delta t$ is small enough that more than one death is highly unlikely.

We can rearrange Eq. (13) to obtain:

\frac{P(n=N;t+\Delta t) - P(n=N;t)}{\Delta t} = P(n=N+1;t)\mu(N+1) - P(n=N;t)\mu N.

(14)

Taking the limit as the time interval $\Delta t$ approaches 0 yields the following differential equation governing the time evolution of the probability $P(n=N;t)$ :

\frac{d P(n=N;t)}{dt} = P(n=N+1;t)\mu(N+1) - P(n=N;t)\mu N.

(15)

Eq. (15) describes the forward Kolmogorov equation for this stochastic death process. It relates the time derivative of the probability of being in state $N$ individuals to the probabilities of transitions between states due to individual death events.

There are several approaches to analyze Eq. (15). One method is to take the time derivative of the expected population size, represented by:

E_n = \sum_{N=0}^{\infty} N P(n=N;t).

(16)

By multiplying both sides of Eq. (15) by N and taking the sum from 0 to infinity, we obtain:

\frac{d}{dt}\sum_{N=0}^{\infty} N P(n=N;t) = \mu \sum_{N=0}^{\infty} N(N+1)P(n=N+1;t) - \mu\sum_{N=0}^{\infty} N^2 P(n=N;t).

(17)

Through some algebraic steps, this equation can be shown to reduce to:

\frac{d E_n}{dt} = - \mu E_n

(18)

Notice that Eq. (18) is identical to the deterministic Eq. (8). This implies the deterministic solution corresponds to the average behavior of repeated stochastic experiments, rather than describing any single experiment. It predicts exponential decay when considering expected population sizes over numerous trials.

Another approach is to find an exact solution to Eq. (15). It can be verified through substitution that:

P(n=N;t) = \frac{E_n^N(t)e^{-E_n(t)}}{N!},

(19)

where $E_n(t)$ , satisfies Eq. (18). This has the form of a Poisson distribution, with mean $E_n$ and standard deviation $\sqrt{E_n}$ .

For large values of $E_n$ , the standard deviation becomes negligible compared to the mean. Therefore, even though the deterministic model in Eq. (8) does not describe individual stochastic trajectories, its predictions are expected to closely match the behavior of single realizations when population sizes are sufficiently large.

While stochastic fluctuations are prominent for small populations, the deterministic exponential decay approximation becomes increasingly accurate as the number of individuals grows. This solution helps connect the statistical properties of the underlying stochastic process to the emergent deterministic behavior predicted by the differential equation model.

Individual Lifetime Distribution¶

As individual death is a stochastic process, the survival time of each individual, denoted by the random variable $\tau$ , is also random. Let $P(\tau=T)dT$ represent the probability that an individual survives up to age $T$ and dies between ages $T$ and $T+dT$ . We can write an equation for this probability as:

P(\tau=T)dT = \left(1 - \int_0^T P(\tau=T')dT' \right)\mu dT.

(20)

The term in parentheses is the probability of surviving until age $T$ , while $\mu dT$ is the probability of dying between $T$ and $T+dT$ .

Differentiating this expression gives the differential equation:

\frac{dP(\tau=T)}{dT} = -\mu P(\tau=T).

(21)

Subject to the normalization condition, the solution is:

P(\tau=T) = \mu e^{-\mu T}.

(22)

Therefore, individual lifetimes follow an exponential distribution. The mean lifetime is thus:

E_\tau = \frac{1}{\mu}.

(23)

Discussion¶

So far, we have analyzed a simple decaying population model considering both deterministic and stochastic frameworks. The key deterministic result was that when death rates are constant over time, the total population size decays exponentially according to Eq. (8). By developing a stochastic description, we were able to account for randomness at the individual level. Interestingly, when considering expected population sizes, the deterministic and stochastic descriptions coincide as shown by Eq. (18).

This emergence of deterministic behavior from underlying stochastic processes is an important phenomenon. While survival of individuals is inherently random, averaging over many trials washes out variability, resulting in smooth exponential decay. This demonstrates how populations self-organize simpler collective dynamics from complex interactions between constituents.

The death rate parameter $\mu$ takes on different meanings depending on the description. Deterministically, it characterizes the system’s exponential decay profile. Stochastically, it represents the probability of individual mortality. Such multi-scale modeling allows $\mu$ to provide insight across descriptive levels.

In summary, even simple population models like exponential decay showcase the interplay between stochastic and deterministic perspectives. Randomness at the individual scale shapes probabilistic population fluctuations, yet deterministic laws emerge at larger scales where variation averages out. This theory establishes foundations for more realistic extensions incorporating additional biological complexities.

Epilogue¶

Let us reexamine the population distribution function $\eta(\tau,t)$ . We can define the normalized function:

\rho(\tau,t) = \frac{\eta(\tau,t)}{N(t)},

(24)

where $N(t) = \int_0^\infty \eta(\tau',t) d\tau'$ . Function $\rho(\tau,t)$ describes the probability density that a randomly selected individual from the population has age $\tau$ at time $t$ .

Rewriting Eq. (6) in terms of $\rho$ :

\frac{\partial\eta}{\partial t} = -\frac{\partial\eta}{\partial\tau} - N(t)\mu(\tau)\rho(\tau,t).

(25)

Upon integration, this leads to:

\frac{dN(t)}{dt} = -\widetilde{\mu}(t)N(t),

(26)

where $\widetilde{\mu}(t) = \int_0^\infty \mu(\tau')\rho(\tau',t)d\tau'$ is the average death rate.

Notably, if the age distribution $\rho(\tau,t)$ remains stationary (independent of t), then the average death rate $\widetilde{\mu}$ is constant. In this case, Eq. (26) again predicts exponential decay of $N(t)$ , even with age-dependent mortality $\mu(\tau)$ .

In conclusion, a stationary age structure is another way of producing exponential population decay through a constant average death rate. Therefore, observing exponential decay alone does not fully characterize the underlying stochastic process. Additional information, such as the lifetime distribution of individuals, is needed to distinguish the underlying stochastic process.