Week 11 - introduction to modelling in biology

We will assume that our population has a simple annual cycle, with all of the following events occurring during each year (and in the following order): 1. Census, during which we assess the number of individuals at the beginning of the year 2. Reproduction during a breeding season that occurs early in the year 3. Mortality, which occurs after the breeding season (e.g., due to a period of intense predation, parasitism, or other seasonal environmental stress) 4. Migration into the population, which occurs towards the end of the year (we assume there is no migration out of the population).

Our model will describe change in population size over time, with population size representing our variable. Let n represent the population size at census in a given year, and n’ represent the population size at census in the following year. Thus, over a single year interval, our variable changes from n to n’. The value of n relative to n’ tells us whether the population size has increased (n’ > n) or decreased (n’ < n).

b is the average number of offspring produced by each individual during the breeding season (i.e., if n individuals reach the breeding season, then we expect nb new offspring produced by the end of the breeding season) · d is the proportion of individuals that die during the period of mortality during the annual cycle (e.g., if 1000 individuals reach the mortality season, we expect 1000d individuals to die and 1000(1 – d) individuals to survive) · m is the number of new migrants entering our population at the end of the year (we assume that the number of entering migrants is independent of the population’s size)

We can visualize the population size throughout an entire year using the following life-cycle diagram, which shows the number of individuals in the population after each event during the year, i.e.: 1. At census (n) 2. After reproduction (nb) 3. After mortality (nd) 4. After migration (nm)

The following train of logic allows leads us to a recursion for our model: · If we start with n individuals at census within a given year, and each of individual has b offspring, then the population size will increase by nb individuals. The total population size after the breeding season becomes: nb = n + nb = n(1 + b). · Immediately prior to the mortality event, we have nb individuals in the population, and d represents the proportion that die (1 – d is the proportion that survives). The population size immediately after the mortality event becomes: nd = nb(1 – d) = n(1 + b)(1 – d). · The final event to occur during the annual cycle is migration, which increases the population size by m (the number of new migrants entering our population), and brings the total populations size to: nm = nd + m = n(1 + b)(1 – d) + m. · Since migration is the final event to occur within a single annual cycle, the population census size at the beginning of the next year (n’) will be equal to the population’s size at the close of the current year, i.e.: n’ = nm = n(1 + b)(1 – d) + m.

n’ = n(1+b)(1-d) + m

Testing verbal intuition (is the idea logical?) Generating new predictions Guiding experimental design and interpretation Establishing formal links between cause and effect

Describe how a variable or variables change over time Population models Gene frequency change in a population Species competition

Deterministic (smoother, dotted line) or stochastic (more random, solid line) Behave pseudo-deterministically when deterministic processes are strong (eg. Strong natural selection vs. weak genetic drift) Discrete time model (steps) or continuous (smooth) Similar dynamics when rate of change is slow Dramatic fluctuation in population size within discrete time but not continuous time

Start by formulating a question eg. What conditions? What timescale or rate? When should we expect stability or instability? Identify variables (things that change over time such as n) and parameters (constants in the model such as b) Qualitatively describe the system - if order of events is important, use discrete time. Construct a life-cycle diagram of events: census, reproduction, mortality and migration Quantitatively describe the system - recursion equations of discrete-time models Analyse the equations eg. What is required for n’ > n? What is the condition for local growth when m=0? Condition for local extinction when m = 0?