Table of Contents
Overview
Population models use differential equations to describe how the size of a group of living organisms changes over time. Typical questions include:
- How quickly does a population grow?
- Does it grow without bound, or level off?
- Under what conditions can it die out?
In this chapter, we focus on simple but important models: exponential growth/decay, logistic growth, and a brief look at interacting populations. The emphasis is on formulating the differential equation, interpreting its terms, and understanding the qualitative behavior of solutions, not on advanced solution techniques.
Throughout, we usually let:
- $t$ = time (e.g., in years)
- $P(t)$ = population size at time $t$ (often assumed positive)
- Parameters like $k$, $r$, $K$ = constants describing how the population changes
Exponential Growth and Decay
Basic idea and differential equation
The simplest assumption is that the rate of change of the population is proportional to the population itself:
$$
\frac{dP}{dt} = k P(t),
$$
where $k$ is a constant:
- If $k > 0$, we have exponential growth.
- If $k < 0$, we have exponential decay.
Interpretation: each individual has the same average chance per unit time of producing new individuals (or dying), so more individuals means a larger total rate of change.
Solution form and interpretation
Solving $\dfrac{dP}{dt} = k P$ with initial condition $P(0) = P_0$ gives
$$
P(t) = P_0 e^{kt}.
$$
Key features:
- If $k > 0$, $P(t)$ increases without bound as $t \to \infty$.
- If $k < 0$, $P(t) \to 0$ as $t \to \infty$.
- The parameter $k$ determines how fast the population changes.
A useful quantity is the doubling time (for $k > 0$):
- Doubling time $T$ satisfies $P(T) = 2P_0$:
$$
2P_0 = P_0 e^{kT} \quad\Rightarrow\quad 2 = e^{kT} \quad\Rightarrow\quad T = \frac{\ln 2}{k}.
$$
Similarly, for decay ($k < 0$), one often uses the half-life $T_{1/2}$, defined by $P(T_{1/2}) = \frac12 P_0$:
$$
T_{1/2} = \frac{\ln 2}{|k|}.
$$
When exponential models are reasonable
Exponential models can approximate population behavior when:
- Resources are effectively unlimited (at least over the time interval considered).
- The environment does not change significantly.
- Birth and death rates per individual are (approximately) constant.
They are usually good over short time spans or for small populations in a very rich environment, but they become unrealistic in the long term because no real population can grow forever without limits.
Logistic Growth
Motivation: limited resources
Real populations usually face:
- Limited food, space, or other resources
- Competition between individuals
- Disease, predation, etc.
These factors tend to slow down growth as the population becomes large. A simple way to model this is to assume that:
- For small $P$, the population grows almost exponentially.
- As $P$ approaches some maximum sustainable size, growth slows and eventually stops.
This leads to the logistic model.
Logistic differential equation
The logistic model is usually written as
$$
\frac{dP}{dt} = r P\left(1 - \frac{P}{K}\right),
$$
where:
- $P(t)$ = population size at time $t$.
- $r > 0$ = intrinsic growth rate (similar to $k$ in the exponential model).
- $K > 0$ = carrying capacity; the maximum sustainable population size given resources.
Interpretations:
- When $P$ is very small compared to $K$, $\frac{P}{K} \approx 0$, so
$$
\frac{dP}{dt} \approx rP,
$$
i.e., nearly exponential growth. - When $P$ is close to $K$, $\frac{P}{K}\approx 1$, so
$$
\frac{dP}{dt} \approx 0,
$$
meaning the population stabilizes. - When $P > K$, the term $\left(1 - \frac{P}{K}\right)$ is negative, so the population tends to decrease back toward $K$.
Equilibria and stability
Equilibria are constant solutions where $\frac{dP}{dt} = 0$.
For the logistic equation:
$$
\frac{dP}{dt} = r P\left(1 - \frac{P}{K}\right) = 0
$$
This happens if:
- $P = 0$, or
- $1 - \frac{P}{K} = 0 \Rightarrow P = K$.
Thus there are two equilibrium populations: $P = 0$ and $P = K$.
To understand stability qualitatively:
- If $0<P<K$, then $P>0$ and $1-\frac{P}{K}>0$, so $\frac{dP}{dt}>0$ and the population increases toward $K$.
- If $P>K$, then $P>0$ and $1-\frac{P}{K}<0$, so $\frac{dP}{dt}<0$ and the population decreases toward $K$.
- If $P<0$ is not biologically meaningful, but mathematically we would see negative values move away from $0$ as well.
So:
- $P = K$ is a stable equilibrium (solutions nearby tend to move toward it).
- $P = 0$ is an unstable equilibrium for $P>0$ (any small positive population will grow away from zero).
Solution form and graph shape (logistic curve)
Solving the logistic equation with initial condition $P(0)=P_0$ gives a solution of the form
$$
P(t) = \frac{K}{1 + A e^{-rt}},
$$
where $A$ is a constant determined by $P_0$:
$$
A = \frac{K - P_0}{P_0}.
$$
Important qualitative features:
- If $0<P_0<K$, then $P(t)$:
- Increases from $P_0$,
- Is always between $0$ and $K$,
- Approaches $K$ as $t \to \infty$.
- The graph is S-shaped (sigmoidal):
- Initially grows almost exponentially,
- Then slows down as it approaches $K$,
- Has a point of fastest growth (inflection point) at $P = \frac{K}{2}$.
Population biologists often fit this kind of curve to data, estimating $r$ and $K$ from observed population sizes over time.
Comparing Exponential and Logistic Models
- Exponential:
$$
\frac{dP}{dt} = kP
$$ - No upper bound on population.
- Growth rate $\frac{dP}{dt}$ is always proportional to $P$, independent of size limits.
- Good model only when resources are effectively unlimited, usually for early-time behavior.
- Logistic:
$$
\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)
$$ - Has a built-in carrying capacity $K$ that population approaches.
- Growth rate decreases as $P$ approaches $K$.
- Often a more realistic long-term model for populations in a fixed environment.
Choosing between them:
- If empirical data show roughly straight-line behavior when plotting $\ln P$ against $t$, exponential might be reasonable.
- If data show leveling off toward a maximum, logistic is more appropriate.
Harvesting and Other Modifications
Real populations may be affected by external actions, such as harvesting (fishing, hunting, logging) or restocking. These effects can be added to basic models.
Constant harvesting in a logistic model
A common modification is to introduce a constant harvest rate $h$, so the logistic model becomes
$$
\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) - h,
$$
where $h \ge 0$ represents the number of individuals removed per unit time.
Key features:
- If $h$ is small, there may still be a positive stable equilibrium population.
- If $h$ is too large, it may be impossible for the population to maintain itself, leading eventually to extinction (population going to $0$).
To find equilibria, solve
$$
rP\left(1 - \frac{P}{K}\right) - h = 0.
$$
This is a quadratic equation in $P$; depending on $h$, it can have:
- Two positive equilibria (one stable, one unstable),
- One positive equilibrium (critical harvesting level),
- No positive equilibrium (overharvesting; long-term extinction).
This kind of analysis is important in resource management (e.g., sustainable fishing quotas).
Other possible terms
More complicated models might include:
- Time-dependent parameters (e.g., seasonal births or deaths).
- Density-dependent death terms that are more complicated than the simple logistic factor.
- Immigration or emigration (individuals entering or leaving the population).
In each case, the added terms modify the differential equation, and we analyze how they affect equilibria and long-term behavior.
Interacting Populations (Brief Overview)
Beyond single populations, differential equations are used to model interactions between species or groups. Two classic examples:
Predator–prey (Lotka–Volterra type)
Let:
- $x(t)$ = prey population (e.g., rabbits),
- $y(t)$ = predator population (e.g., foxes).
A simple model might be
$$
\begin{aligned}
\frac{dx}{dt} &= ax - bxy, \\
\frac{dy}{dt} &= -cy + dxy,
\end{aligned}
$$
with positive constants $a,b,c,d$.
Interpretation (at a basic level):
- Prey grow exponentially ($ax$) in absence of predators, but are eaten at rate proportional to both populations ($bxy$).
- Predators die out without prey ($-cy$), but increase in presence of prey ($dxy$).
This system can produce oscillations: prey numbers rise, then predators increase and reduce prey, leading predators to fall, and so on.
Competing species
If two species compete for the same resources, both populations may be modeled with logistic-type terms that also depend on the other species. For instance, one can add terms that reduce each species’ growth in proportion to the size of the other.
The precise form depends on the situation and empirical data, but the general idea remains: the rate of change depends on both the species’ own population and its interactions with others.
Using Population Models in Practice
In applications, the steps are usually:
- Model formulation
- Choose variables (e.g., $P(t)$).
- Decide which factors to include (growth rate, carrying capacity, harvesting, interactions).
- Write a differential equation that reflects these assumptions.
- Parameter estimation
- Use data (population counts over time, birth/death rates, etc.) to estimate parameters like $k$, $r$, $K$, $h$.
- Often done with statistical or numerical methods.
- Analysis
- Identify equilibria.
- Determine stability (qualitative behavior near equilibria).
- Study long-term trends (extinction, stabilization at carrying capacity, unbounded growth, oscillations).
- Interpretation
- Use results to inform decisions (e.g., setting harvest quotas, conservation strategies).
- Check whether model predictions match further data; if not, refine the model.
Population models are simplifications, but even the basic differential equations discussed here capture important qualitative behaviors: exponential explosion, saturation at a carrying capacity, possible extinction under heavy harvesting, and complex dynamics when multiple species interact.