Piret Avila, Nov 5 2025

Last week (Population Dynamics)

We only looked at lumped-parameter models, where all individuals have identical demographic patters (survival, reproductive output)

Now, we are introducing fitness as a growth rate of the population $r$, the Malthusian parameter, measured on expentional scale

• Over one demographic time step, we denote fitness = lambda $\lambda$- this comes from matrix algebra
• When $r>0$, population is growing ($w=\lambda >1$)
• When $r<0$, population is declining ($w=\lambda <1$)

Class-structured Models

These models go beyond this assumption. We will introduce that individuals in the population belong to classes. Each class has its own demogrpahic characteristics. These could include:

• age
• size
• sex
• developmental stage
• physiological condition

Contiguous vs discrete

Some variables like physiological state are distributed continuously, while some are discrete categories. Age can be thought of as continuous or put into categories. We are treating all classes as discrete by putting them into categories.

Central Quesion

Can we predict population dynamics from just counting individuals?

No, we need to consider demogrpahics.

Euler first described this in terms of how mortality depends on age - we'll come back to this later.

Life cycles / Life Histories

• Major events related to birth, growth , development, reproduction and survival
• Fundamental unit of description in the organism
  • Link between individual and population
• Examples of Life History Adaptations:
  • Age and size at sexual maturity
  • Amount and timing of reproduction
  • Survival and mortality rates

Life Tables

These can be used to summarize how reproductive rates vary with age

There are two types:

• Period Life Tables
• Cohort Life Tables

Period Life tables

• Also known as static life tables or time-specific
• Data is affected by period events that affect many individuals like wars, pandemic or famine
• Example: Life expentancy at birth, life expectancy for an 80 year old

Cohort life table

• Also known as dynamic life tables or horizontal life tables
• Calculated based on a cohort of individuals born during the same period
• Affected by cohort effects that happened at a particular time but carried on as the group ages
  • e.g. being born during a pandemic or famine will make your cohort have a lower life expectancy

Cohort life table example: kylänurmikka

Annual bluegrass (Poa annua) shows extreme plasticity and rapid adaptation

Meticulous calculations for life tables

Projection equations

Leslie's innovation

Patrick Holt Leslie in 1945 determined taht you can use matrix algebra and a projection matrix to avoid all these calculations

Leslie Matrix

Where $A$ is the Leslie matrix, build from the life table. This is similar to our previous equation $N_{t+1}=w(n)n_t$, but using the matrix makes this more complicated by incorporating the life tbale matrix

!Lecture_2_Handout, p.24

We get the survival probabilities from our life table, from survivorship at time x+1 divided by survivorship at time x.
$s_x=S_x=I_{x+1}/I_x$

And the Fertility rate values is fecundity at time x+1 times survival rate at time x
$f_x=F_{x+1}\cdot S_x$
(Accounts for survival to reproduction and births produced)

Long-term effects of matrix math:

Eventuall, we find out the eigenvalue or point where the population grows or declines at a constant rate $\lambda$ and achieves a stable age distribution:

$\lambda$ is the finite rate of increase - multiplative change in population size per time step.