State-Dependent Birth And Death Rates

Rates Describe Local Changes

In a population model, the chance of a birth or death during the next short time interval can depend on the current population size. A model with this feature is a birth-death process: at each jump, the state changes by one, either upward through a birth or downward through a death. The rates record how quickly these changes can occur at each state. They are not probabilities, and their values need not be the same from one state to another.

Work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. A process $(X_t)_{t\ge0}$ with state space $S=\{0,1,2,\ldots\}$ records the population or count at time $t$. Assume it is a time-homogeneous continuous-time Markov chain: given its current state, its future transition behavior does not depend on its earlier history or on the absolute time. Write $\mathbb{P}_n$ for probabilities when $X_0=n$. The rates below specify its local transition behavior whenever such a process is defined.

Definition: A birth-death process on $S$ has birth rates $\lambda_n\ge0$ and death rates $\mu_n\ge0$, with $\mu_0=0$. From state $n$, its only possible one-step jumps are to $n+1$, at rate $\lambda_n$, and, if $n\ge1$, to $n-1$, at rate $\mu_n$. For a time-homogeneous chain, the local rates are expressed as $$ \mathbb{P}_n(X_h=n+1)=\lambda_n h+o(h),\qquad \mathbb{P}_n(X_h=n-1)=\mu_n h+o(h) $$ as $h\downarrow0$, with the second equation applying only when $n\ge1$. The probability of no change over this short interval is $1-(\lambda_n+\mu_n)h+o(h)$; the probability of being at any other state is $o(h)$. At $n=0$, use $\mu_0=0$, so the displayed no-change expression is $1-\lambda_0h+o(h)$.

Here $o(h)$ denotes a term whose ratio to $h$ tends to zero as $h\downarrow0$. The short-time expressions distinguish a rate from a probability: for example, $\lambda_n$ has units of inverse time, whereas $\lambda_n h$ is a first-order approximation to a probability. Rates may exceed one. The expressions are local approximations, not exact formulas for an interval of arbitrary length.

The death rate at state $0$ must be zero because a downward jump would leave the state space. A zero birth rate $\lambda_n=0$ means there is no upward arrow from $n$; a zero death rate $\mu_n=0$ means there is no downward arrow. Such missing arrows can arise at states other than zero too. When $\lambda_n+\mu_n=0$, the local specification gives no outgoing jump from $n$. In a standard birth-death chain, that state is absorbing.

From Rates to Allowed Jumps

A rate diagram puts the states in order and labels the arrows between neighbors. The arrow from $n$ to $n+1$ is labelled $\lambda_n$, and the arrow from $n$ to $n-1$ is labelled $\mu_n$. Each arrow is directed: the rate for moving upward from $n$ need not equal the rate for moving downward from $n+1$. A diagram can therefore display the same pair of neighboring states with different rates in opposite directions.

0 \;\underset{\mu_1}{\overset{\lambda_0}{\rightleftarrows}}\; 1 \;\underset{\mu_2}{\overset{\lambda_1}{\rightleftarrows}}\; 2 \;\underset{\mu_3}{\overset{\lambda_2}{\rightleftarrows}}\; 3 \;\underset{\mu_4}{\overset{\lambda_3}{\rightleftarrows}}\;\cdots

The leftward arrow from $0$ is absent, not merely unlabelled: $\mu_0=0$. If another rate is zero, its arrow is absent as well. No arrow skips a state. Thus, for example, the diagram does not include a direct jump from $n$ to $n+2$; reaching $n+2$ from $n$ requires at least two successive jumps.

Proposition: The local rate specification of a birth-death process determines exactly the possible one-step jump destinations and their rates: from $n$, only neighboring states can be reached in one jump, and the rates on those arrows are $\lambda_n$ and $\mu_n$. Conversely, a rate diagram with only these neighboring arrows, labelled by nonnegative rates and with no downward arrow from $0$, specifies the birth and death rates.

Proof. By the definition, a birth from $n$ changes the state to $n+1$ and has rate $\lambda_n$; a death changes it to $n-1$ and has rate $\mu_n$. The definition permits no other one-step jump, so there are no other arrows. At state $0$, the condition $\mu_0=0$ excludes a downward arrow. Conversely, given a diagram of the stated form, define $\lambda_n$ to be the label on the arrow from $n$ to $n+1$, using zero if that arrow is absent, and define $\mu_n$ to be the label on the arrow from $n$ to $n-1$, using zero if absent. The boundary condition sets $\mu_0=0$. These assignments give precisely the specified neighboring jump rates and no others, as required. $\square$

A rate diagram records local transition information; it does not by itself establish every global property of a process. In particular, for an infinite state space, arbitrary finite rates at each individual state do not automatically guarantee that a process can be continued for all time without infinitely many jumps occurring in a finite interval. That issue is called non-explosion. It is separate from reading the allowed jumps. The examples below illustrate the local specifications and use bounded rates.

Reading and Checking Rate Diagrams

Worked Example: Constant Birth and Death Rates Above Zero

Consider rates $\lambda_n=2$ for every $n\ge0$, $\mu_0=0$, and $\mu_n=1$ for every $n\ge1$. Find the possible one-step jumps and the combined local rate $\lambda_n+\mu_n$ at states $0$, $1$, and $2$.

At state $0$, the birth rate is $\lambda_0=2$ and the death rate is $\mu_0=0$. Thus the only outgoing arrow is $0\to1$, labelled $2$, and the combined local rate is $2+0=2$.

At state $1$, $\lambda_1=2$ gives the arrow $1\to2$, while $\mu_1=1$ gives $1\to0$. The combined local rate is $2+1=3$. At state $2$, $\lambda_2=2$ gives $2\to3$, and $\mu_2=1$ gives $2\to1$, again with combined local rate $2+1=3$.

0 \;\xrightarrow{\;2\;}\; 1 \;\underset{\;1\;}{\overset{\;2\;}{\rightleftarrows}}\; 2 \;\underset{\;1\;}{\overset{\;2\;}{\rightleftarrows}}\; 3 \;\underset{\;1\;}{\overset{\;2\;}{\rightleftarrows}}\;\cdots

Check: At $0$, the missing leftward arrow agrees with $\mu_0=0$. At $1$ and $2$, the upward labels are both $2$, and the downward labels are both $1$, exactly as given by the rates. The combined rates computed from the displayed values are $2$ at $0$ and $3$ at each of $1$ and $2$.

Worked Example: A Missing Birth Arrow

Let $\lambda_0=3$, $\lambda_1=2$, $\lambda_2=0$, and $\lambda_n=1$ for $n\ge3$. Set $\mu_0=0$ and $\mu_n=2$ for every $n\ge1$. Determine the arrows and the combined local rates at states $0$ through $3$.

From state $0$, there is a birth arrow to $1$ labelled $3$, and no downward arrow. Its combined local rate is $3+0=3$. From state $1$, the birth arrow to $2$ is labelled $2$, and the death arrow to $0$ is labelled $2$; the combined local rate is $2+2=4$.

At state $2$, the birth rate is zero, so there is no arrow from $2$ to $3$. The death arrow $2\to1$ has rate $2$, giving combined local rate $0+2=2$. At state $3$, the birth arrow $3\to4$ has rate $1$ and the death arrow $3\to2$ has rate $2$; the combined local rate is $1+2=3$.

0 \;\xrightarrow{\;3\;}\; 1 \;\underset{\;2\;}{\overset{\;2\;}{\rightleftarrows}}\; 2 \;\xleftarrow{\;2\;}\; 3 \;\underset{\;2\;}{\overset{\;1\;}{\rightleftarrows}}\; 4

Check: The diagram has no arrow $2\to3$, matching $\lambda_2=0$, but it retains the arrow $3\to2$, matching $\mu_3=2$. The calculated combined local rates, in state order $0,1,2,3$, are $3,4,2,3$, obtained respectively as $3+0$, $2+2$, $0+2$, and $1+2$.

Worked Example: Rates That Vary With the State

Suppose $\lambda_n=\frac{4}{n+1}$ for $n\ge0$, $\mu_0=0$, and $\mu_n=\frac{n}{n+1}$ for $n\ge1$. Compute the birth and death rates, possible jumps, and combined local rates at $0$, $1$, and $2$.

At $0$, substitution gives $\lambda_0=\frac{4}{0+1}=4$ and $\mu_0=0$. There is only the arrow $0\to1$, and the combined local rate is $4+0=4$.

At $1$, substitution gives $\lambda_1=\frac{4}{1+1}=2$ and $\mu_1=\frac{1}{1+1}=\frac12$. Thus the arrows are $1\to2$ at rate $2$ and $1\to0$ at rate $\frac12$. Their combined local rate is $2+\frac12=\frac52$.

At $2$, $\lambda_2=\frac{4}{2+1}=\frac43$ and $\mu_2=\frac{2}{2+1}=\frac23$. The arrows are $2\to3$ at rate $\frac43$ and $2\to1$ at rate $\frac23$, with combined local rate $\frac43+\frac23=\frac63=2$.

Check: Each computed birth rate equals $4/(n+1)$ at its state, and each death rate for $n\ge1$ equals $n/(n+1)$. The combined rate at state $2$ is $2$, since $\frac43+\frac23=2$; at state $1$ it is $\frac52$, and at state $0$ it is $4$. Only neighboring states appear in the diagram.

Common Misreadings

A frequent mistake is to treat a rate as a one-step probability. For example, in the third example the birth rate at state $0$ is $4$. Writing “the probability of a birth in the next unit of time is $4$” is wrong: probabilities cannot exceed $1$, and the rate has units of inverse time. The local statement is instead $\mathbb{P}_0(X_h=1)=4h+o(h)$ as $h\downarrow0$. This approximation is for sufficiently short intervals and is not a formula for a unit interval.

Another error is to put the birth rate $\lambda_n$ on the arrow into $n$, rather than on the arrow out of $n$. The index is the current state: $\lambda_n$ labels $n\to n+1$, and $\mu_n$ labels $n\to n-1$. In particular, the downward arrow between $n$ and $n+1$ has rate $\mu_{n+1}$, not $\mu_n$.

Finally, zero rates should be represented by missing arrows. The example with $\lambda_2=0$ still allows a death from $2$ to $1$, and it allows a birth from $3$ to $4$. A zero rate removes only its own directed transition; it does not remove the state or automatically remove the arrow in the opposite direction.

For any proposed birth-death specification, first check that all rates are nonnegative and finite at each state, that $\mu_0=0$, and that the arrow from $n$ to $n+1$ is labelled $\lambda_n$ while the reverse arrow is labelled $\mu_{n+1}$. These checks establish that the local rate diagram matches the stated model.

Check Your Understanding

Use the definitions and rate-diagram conventions to answer the following.

If $\lambda_1=5$ and $\mu_1=2$, which states can the process reach in one jump from state $1$, and what labels appear on the corresponding arrows?
Suppose $\lambda_2=0$ but $\mu_2>0$. Which directed arrow is absent, and which may still be present?
For $\lambda_n=3$ at every state and $\mu_0=0,\ \mu_n=1$ for $n\ge1$, calculate the combined local rates at states $0$ and $2$.
Explain why a birth rate is not itself the probability of a birth during an arbitrary time interval.
What boundary condition on the death rates is required for a process whose state space is the nonnegative integers, and what does it mean on the rate diagram?

State-Dependent Birth And Death Rates

What You'll Learn