The Central Idea
A population process describes how the number or state of individuals changes over time.
The simplest model tracks only population size:
\[ N_0,N_1,N_2,\ldots \]More detailed models divide the population into classes, such as:
- juvenile and adult;
- susceptible and infected;
- alive and dead;
- employed and unemployed;
- location A and location B.
The central mathematical question is:
\[ \boxed{\text{how does the population state change from one time step to the next?}} \]Population Balance
A basic population accounting identity is
\[ \boxed{ N_{\text{new}} = N_{\text{old}} +B-D+I-E }. \]Here:
- \(B\) is births;
- \(D\) is deaths;
- \(I\) is immigration;
- \(E\) is emigration.
A First Population Example
Suppose a population begins the year with
\[ 1200 \]individuals.
During the year:
- \(180\) are born;
- \(90\) die;
- \(40\) immigrate;
- \(25\) emigrate.
Then
\[ 1200+180-90+40-25 = \boxed{1305}. \]Discrete-Time Growth
A simple deterministic growth model assumes that population changes by a fixed proportion each time step.
If \(r\) is the per-period growth rate, then
\[ \boxed{ N_{t+1}=(1+r)N_t }. \]Let
\[ \lambda=1+r. \]Then
\[ \boxed{N_{t+1}=\lambda N_t}. \]Geometric Population Growth
Repeated application gives
\[ N_t=N_0\lambda^t. \]Therefore,
\[ \boxed{ N_t=N_0(1+r)^t }. \]This is the discrete analogue of exponential growth.
Growth Example
Suppose
\[ N_0=500 \]and the population increases by \(8\%\) per year.
Then
\[ \lambda=1.08. \]After \(4\) years,
\[ N_4=500(1.08)^4 \approx \boxed{680.2}. \]A real count would ordinarily be interpreted with appropriate rounding or a more detailed integer-valued model.
Growth Factor Interpretation
If
\[ \lambda>1, \]the population grows geometrically.
If
\[ \lambda=1, \]the population remains constant.
If
\[ 0<\lambda<1, \]the population declines geometrically.
Per-Capita Change
A useful quantity is change per individual.
If
\[ \Delta N=N_{t+1}-N_t, \]then the proportional change is
\[ \boxed{ \frac{\Delta N}{N_t} }. \]In the simple geometric model, this quantity equals \(r\).
Birth and Death Rates
Suppose each individual contributes an average birth rate \(b\) and death rate \(d\) per period.
A simple deterministic recurrence is
\[ N_{t+1} = N_t+bN_t-dN_t. \]Therefore,
\[ \boxed{ N_{t+1}=(1+b-d)N_t }. \]Net Growth Rate
Define
\[ r=b-d. \]Then
\[ N_{t+1}=(1+r)N_t. \]Birth and death rates therefore combine into a net growth rate in the simplest model.
Density Dependence
Unlimited geometric growth is often unrealistic.
As population grows, competition for:
- food;
- space;
- nesting sites;
- water;
- other resources
may reduce future growth.
This motivates density-dependent models.
The Logistic Map
One discrete density-dependent model is
\[ \boxed{ N_{t+1} = N_t+rN_t\left(1-\frac{N_t}{K}\right) }. \]Here:
- \(r\) controls growth strength;
- \(K\) is a carrying-capacity parameter.
Carrying Capacity
The parameter \(K\) represents a population scale at which net growth is reduced to zero in the standard logistic form.
When
\[ N_t=K, \]then
\[ 1-\frac{N_t}{K}=0. \]Therefore,
\[ N_{t+1}=N_t. \]Logistic Example
Suppose
\[ N_t=400, \qquad r=0.3, \qquad K=1000. \]Then
\[ N_{t+1} = 400+0.3(400)\left(1-\frac{400}{1000}\right). \]Since
\[ 1-\frac{400}{1000}=0.6, \]we get
\[ N_{t+1} = 400+120(0.6) = 400+72 = \boxed{472}. \]Deterministic Versus Stochastic Models
A deterministic model gives one next state from a given current state.
A stochastic model gives a probability distribution over possible next states.
For example, instead of assuming exactly \(20\) births, a model may let the number of births be random.
Why Randomness Matters in Populations
Random variation can arise from:
- which individuals reproduce;
- which individuals survive;
- weather;
- disease;
- migration;
- resource fluctuations;
- random encounters.
Randomness is especially important in small populations.
Binomial Survival Model
Suppose a population has \(N\) individuals and each survives independently with probability \(s\).
Then the number of survivors \(X\) can be modeled as
\[ \boxed{ X\sim\operatorname{Binomial}(N,s) }. \]Therefore,
\[ E[X]=Ns. \]Survival Example
Suppose
\[ N=100 \]and each individual survives independently with probability
\[ s=0.85. \]Then the expected number of survivors is
\[ E[X]=100(0.85)=\boxed{85}. \]Expected Value Is Not a Guarantee
If the expected number of survivors is \(85\), the actual number need not be exactly \(85\).
Expected value is a long-run average or probability-weighted center, not a guaranteed outcome for one population.
Branching Processes
A branching process models reproduction when each individual independently produces a random number of descendants.
Let
\[ X \]be the number of offspring produced by one individual.
If the current generation contains \(Z_t\) individuals, then
\[ \boxed{ Z_{t+1} = X_1+X_2+\cdots+X_{Z_t} }. \]Mean Offspring Number
Let
\[ m=E[X]. \]Then \(m\) is the expected number of descendants per individual.
In a basic Galton-Watson branching process:
- \(m<1\) is subcritical;
- \(m=1\) is critical;
- \(m>1\) is supercritical.
Expected Branching Population
If
\[ Z_0=z_0 \]and each individual has mean offspring number \(m\), then
\[ \boxed{ E[Z_t]=z_0m^t }. \]This resembles geometric growth, but the actual population remains random.
Branching Example
Suppose
\[ Z_0=10 \]and
\[ m=1.2. \]Then
\[ E[Z_3] = 10(1.2)^3 = 10(1.728) = \boxed{17.28}. \]Extinction
In a branching process, extinction occurs if the population eventually reaches
\[ Z_t=0. \]Once the process reaches zero, no further descendants appear in the basic model.
Supercritical Does Not Mean Guaranteed Survival
Even when
\[ m>1, \]extinction can still occur with positive probability in many branching processes.
The condition \(m>1\) concerns expected reproduction, not certainty for one realization.
Generating Functions in Branching Processes
If offspring probabilities are
\[ P(X=k)=p_k, \]the probability generating function is
\[ \boxed{ G(s)=\sum_{k=0}^{\infty}p_ks^k }. \]Generating functions provide a compact way to study offspring distributions and extinction probabilities.
Extinction Equation
In a standard branching process, the eventual extinction probability \(q\) is the smallest solution in \([0,1]\) of
\[ \boxed{G(q)=q}. \]This is an important example of a probability problem becoming a fixed-point problem.
Age-Structured Populations
Populations often contain classes with different survival and reproduction rates.
For example, we may divide a population into:
\[ \text{juveniles},\qquad \text{adults}. \]A vector can represent the state:
\[ \boxed{ \mathbf n_t= \begin{bmatrix} J_t\\ A_t \end{bmatrix} }. \]Matrix Population Models
A linear class-structured model can be written
\[ \boxed{ \mathbf n_{t+1}=A\mathbf n_t }. \]The entries of \(A\) can represent:
- fertility;
- survival;
- transition between classes.
A Two-Class Example
Suppose
\[ A= \begin{bmatrix} 0 & 1.5\\ 0.6 & 0.8 \end{bmatrix} \]and
\[ \mathbf n_t= \begin{bmatrix} 100\\ 200 \end{bmatrix}. \]Then
\[ \mathbf n_{t+1} = \begin{bmatrix} 0(100)+1.5(200)\\ 0.6(100)+0.8(200) \end{bmatrix}. \]Therefore,
\[ \boxed{ \mathbf n_{t+1} = \begin{bmatrix} 300\\ 220 \end{bmatrix} }. \]Markov Population States
Sometimes the number of individuals is fixed while individuals move among states.
Examples include:
- healthy, sick, recovered;
- habitat A, B, or C;
- employed or unemployed;
- juvenile or adult class;
- susceptible or immune.
Markov chains can model these transitions.
Transition Matrices
Suppose a two-state system has transition matrix
\[ P= \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix}. \]Each row gives probabilities of moving from one current state to the next state.
Each row sums to
\[ \boxed{1}. \]Population Proportions Under Markov Transitions
If the row vector
\[ \boldsymbol\pi_t \]gives current state proportions, then
\[ \boxed{ \boldsymbol\pi_{t+1} = \boldsymbol\pi_tP }. \]Repeated multiplication models the evolution of the population distribution across states.
Migration Models
Population change may occur through migration between locations.
A transition matrix can represent the fraction moving from one region to another each period.
This can model:
- animal movement;
- human migration;
- cell-state transitions;
- movement among habitats.
Survival Analysis as State Transition
A simple survival process has states
\[ \text{alive} \]and
\[ \text{dead}. \]Death is an absorbing state because once entered, the process does not return to alive in the standard model.
Absorbing States
A state \(i\) is absorbing if
\[ \boxed{P_{ii}=1}. \]Once the process enters that state, it remains there.
Absorbing-state models appear in:
- death;
- extinction;
- recovery without reinfection in simplified models;
- completion of a biological stage.
Infection Processes
Disease spread can be modeled by dividing a population into health states.
A classic structure is:
\[ S=\text{susceptible}, \qquad I=\text{infectious}, \qquad R=\text{removed or recovered}. \]The state at time \(t\) can be written
\[ (S_t,I_t,R_t). \]Discrete SIR-Style Change
A simplified discrete-time model might use
\[ \Delta S_t = -\beta\frac{S_tI_t}{N}, \] \[ \Delta I_t = \beta\frac{S_tI_t}{N} - \gamma I_t, \] \[ \Delta R_t = \gamma I_t. \]This is a deterministic approximation. Real infection and recovery counts can be modeled stochastically.
Basic Reproduction Number
In epidemic modeling, a reproduction measure such as
\[ R_0 \]summarizes the expected number of secondary infections generated by a typical infectious individual under specified conditions.
The exact formula depends on the model.
In simple settings:
- \(R_0>1\) suggests potential growth of infection;
- \(R_0<1\) suggests decline.
Expected Spread Is Not Guaranteed Spread
Even if a reproduction measure exceeds \(1\), early transmission can still die out by chance when case numbers are small.
This is another branching-process idea:
\[ \boxed{\text{average growth can coexist with a positive extinction probability}}. \]Contact Networks
Homogeneous mixing assumes every individual interacts in roughly the same way.
Real contact patterns are often networked.
A graph can represent:
- individuals as vertices;
- potential transmission contacts as edges.
Network structure can strongly affect spread.
Population Variance
Stochastic population models describe not only expected population size but also variability.
Two models can have the same
\[ E[N_t] \]while having very different
\[ \operatorname{Var}(N_t). \]Variability matters when extinction, thresholds, or capacity limits are important.
Small-Population Risk
Random fluctuations have greater proportional effects in small populations.
Losing \(10\) individuals from a population of \(10000\) is a small fractional change.
Losing \(10\) from a population of \(20\) is potentially catastrophic.
This is why stochastic models are often especially important in conservation and outbreak initiation.
Demographic Stochasticity
Demographic stochasticity is random variation arising from individual birth, death, reproduction, or transition events.
Even in a constant environment, individual outcomes differ.
Environmental Stochasticity
Environmental stochasticity arises when external conditions affect many individuals simultaneously.
Examples include:
- drought;
- temperature shifts;
- storms;
- resource collapse;
- regional disease pressure.
Environmental effects can create correlated population outcomes.
Population Thresholds
Some systems behave differently above or below a threshold.
Thresholds can arise from:
- minimum breeding population;
- herd-immunity effects;
- resource limits;
- habitat capacity;
- minimum viable group size.
Threshold models are often nonlinear.
Harvesting and Removal
A population may experience planned removal through harvesting, culling, or extraction.
A simple recurrence is
\[ \boxed{ N_{t+1}=f(N_t)-H_t }. \]Here \(f(N_t)\) describes biological growth and \(H_t\) is the number removed.
Harvesting Example
Suppose a population grows according to
\[ N_{t+1}=1.1N_t \]before harvest.
If
\[ N_t=500 \]and \(40\) individuals are removed, then
\[ N_{t+1}=1.1(500)-40. \]Therefore,
\[ N_{t+1}=550-40=\boxed{510}. \]Stochastic Harvest Risk
A harvest level that appears sustainable under average growth can become risky if actual reproduction varies.
Therefore, management decisions may need to consider:
- expected growth;
- variance;
- bad-year probabilities;
- minimum population thresholds;
- uncertainty in parameter estimates.
Simulation of Population Processes
Simulation is useful when:
- transitions are stochastic;
- several population classes interact;
- closed-form solutions are difficult;
- extinction probabilities are important;
- management policies must be compared.
A Simple Survival Simulation
Suppose \(100\) individuals each survive one period with probability
\[ 0.9. \]One simulation can generate \(100\) independent Bernoulli outcomes and count the survivors.
Repeating the simulation many times estimates the distribution of future population size.
Monte Carlo Population Analysis
A Monte Carlo study repeats a stochastic model many times.
It can estimate quantities such as:
- mean population size;
- extinction probability;
- probability of falling below a threshold;
- distribution of outbreak size;
- time to extinction;
- policy risk.
Common Error: Treating Expected Population as the Actual Population
If
\[ E[N_t]=50, \]one realization may produce
\[ 41,\ 53,\ 67, \]or another value.
Expected value summarizes the distribution; it does not determine one sample path.
Common Error: Assuming Constant Percentage Growth Forever
Geometric growth can be useful over limited ranges.
But long-run growth may be limited by:
- resources;
- space;
- competition;
- disease;
- environmental change.
Model assumptions should match the scale and time horizon.
Common Error: Confusing Growth Rate and Growth Factor
If the growth rate is
\[ r=0.08, \]the growth factor is
\[ 1+r=1.08. \]Multiplying by \(0.08\) gives only the change amount, not the next total.
Common Error: Ignoring Integer Counts
Deterministic models can produce values such as
\[ 680.2. \]Real populations contain whole individuals.
Fractional values may represent an expectation, approximation, density, or scaled quantity rather than a literal count.
Common Error: Assuming Independence Without Justification
Individual survival events may be correlated if individuals share the same environment.
A drought, storm, or disease event can affect many individuals together.
Independence is a model assumption, not an automatic fact.
Common Error: Mixing Deterministic and Stochastic Quantities
A recurrence such as
\[ N_{t+1}=1.05N_t \]produces a deterministic next value.
A branching or binomial model produces a probability distribution.
These should not be interpreted in the same way.
Common Error: Treating \(R_0\) as a Fixed Universal Constant
Reproduction measures depend on:
- the model;
- population behavior;
- contact patterns;
- environment;
- immunity;
- interventions.
Their interpretation must remain tied to the assumptions used to estimate them.
A Reliable Procedure for Deterministic Population Models
- Define the population being counted.
- Choose the time step.
- Identify births, deaths, immigration, and emigration.
- Determine whether proportional growth is reasonable.
- Write the recurrence relation.
- Calculate one step carefully.
- Iterate or derive a closed form when possible.
- Check whether resource limits or density dependence should be included.
- Interpret fractional outputs appropriately.
A Reliable Procedure for Stochastic Population Models
- Define the random events.
- Identify the probability distribution for each event.
- State independence or dependence assumptions.
- Write the transition or reproduction rule.
- Calculate expected values and variances when useful.
- Identify absorbing states or extinction conditions.
- Use simulation when the full distribution is difficult to derive.
- Report probabilities, not only averages.
A Reliable Procedure for Markov Population Models
- Define the states.
- Estimate transition probabilities.
- Build the transition matrix.
- Verify that each row sums to \(1\).
- Represent the current state distribution.
- Multiply by the transition matrix.
- Repeat for multiple periods if required.
- Interpret stable or absorbing behavior when relevant.
Example: Full Deterministic Workflow
Suppose a population begins at
\[ N_0=800 \]and grows by \(5\%\) per year.
The growth factor is
\[ \lambda=1.05. \]After \(3\) years,
\[ N_3=800(1.05)^3. \]Therefore,
\[ N_3 = 800(1.157625) = \boxed{926.1}. \]Example: Full Stochastic Survival Workflow
Suppose \(60\) animals each survive independently with probability
\[ s=0.8. \]Then
\[ X\sim\operatorname{Binomial}(60,0.8). \]Expected survivors:
\[ E[X]=60(0.8)=\boxed{48}. \]Variance:
\[ \operatorname{Var}(X) = 60(0.8)(0.2) = \boxed{9.6}. \]Example: Full Markov Workflow
Suppose a population is divided between locations A and B.
Transition matrix:
\[ P= \begin{bmatrix} 0.9 & 0.1\\ 0.2 & 0.8 \end{bmatrix}. \]Current proportions:
\[ \boldsymbol\pi_0= \begin{bmatrix} 0.6 & 0.4 \end{bmatrix}. \]Then
\[ \boldsymbol\pi_1 = \boldsymbol\pi_0P. \]Therefore,
\[ \boldsymbol\pi_1 = \begin{bmatrix} 0.6(0.9)+0.4(0.2) & 0.6(0.1)+0.4(0.8) \end{bmatrix}. \]So
\[ \boxed{ \boldsymbol\pi_1= \begin{bmatrix} 0.62 & 0.38 \end{bmatrix} }. \]Practice 1: Population Balance
A population begins at \(900\), has \(120\) births, \(70\) deaths, \(25\) immigrants, and \(15\) emigrants.
Find the new population.
Practice 2: Growth Factor
A population grows by \(6\%\) per year.
Find the growth factor.
Practice 3: Geometric Growth
A population begins at \(400\) and grows by \(10\%\) per period.
Find the population after \(2\) periods.
Practice 4: Expected Survivors
A group contains \(80\) individuals, each surviving independently with probability \(0.75\).
Find the expected number of survivors.
Practice 5: Branching Expectation
A branching process begins with \(5\) individuals and has mean offspring number
\[ m=1.4. \]Find
\[ E[Z_2]. \]Practice 6: Logistic Update
Suppose
\[ N_t=200,\qquad r=0.5,\qquad K=1000. \]Use
\[ N_{t+1}=N_t+rN_t\left(1-\frac{N_t}{K}\right) \]to find the next population.
Practice 7: Markov Row Check
Is
\[ [0.3,\ 0.4,\ 0.3] \]a valid probability row of a transition matrix?
Practice 8: Absorbing State
What must the self-transition probability \(P_{ii}\) equal for state \(i\) to be absorbing?
Practice 9: Harvesting
A population of \(300\) grows by \(20\%\) before a harvest of \(50\).
Find the population after growth and harvest.
Practice 10: Expected Value Interpretation
If the expected future population is \(72\), must the actual future population equal \(72\)?
Practice Answers
Practice 1
\[ 900+120-70+25-15 = \boxed{960}. \]Practice 2
\[ 1+0.06=\boxed{1.06}. \]Practice 3
\[ N_2=400(1.10)^2 =400(1.21) = \boxed{484}. \]Practice 4
\[ E[X]=80(0.75)=\boxed{60}. \]Practice 5
\[ E[Z_2]=5(1.4)^2 =5(1.96) = \boxed{9.8}. \]Practice 6
\[ N_{t+1} = 200+0.5(200)\left(1-\frac{200}{1000}\right). \] \[ = 200+100(0.8) = \boxed{280}. \]Practice 7
Yes.
\[ 0.3+0.4+0.3=\boxed{1}. \]Practice 8
\[ \boxed{P_{ii}=1}. \]Practice 9
First grow:
\[ 300(1.2)=360. \]Then harvest:
\[ 360-50=\boxed{310}. \]Practice 10
No.
Expected value is a probability-weighted average, not a guaranteed outcome for one realization.
What to Remember
Biological and population processes combine recurrence relations, probability, stochastic processes, matrices, and simulation.
- population balance tracks births, deaths, immigration, and emigration;
- geometric growth uses \(N_t=N_0(1+r)^t\);
- density dependence can limit growth;
- logistic models introduce carrying-capacity effects;
- binomial models can represent independent survival events;
- branching processes model random reproduction;
- supercritical expected growth does not guarantee survival;
- Markov chains model movement among population states;
- absorbing states represent irreversible outcomes in simplified models;
- epidemic processes combine contact, transmission, recovery, and stochastic spread;
- simulation helps estimate extinction, threshold, and outbreak probabilities.
The central habit is: decide whether the system is best modeled by deterministic change, random individual events, state transitions, or a combination of all three.
Where We Go Next
Biological and population processes show how discrete and stochastic models describe growth, survival, transition, and extinction. The next applied problem family uses similar probability tools to study uncertain financial outcomes, claims, losses, portfolios, and risk.
The next lesson, Finance, Insurance, and Risk, develops expected-value, probability, loss-distribution, and decision models for financial and insurance systems.