BMED 3510 Population Models Book Chapter 10

General Questions How does the size of a population change over time? Speed of growth Final population size; carrying capacity; steady state Stability Dynamics if size is above carrying capacity How does the composition of a population change over time? Age structure Dynamic changes in sizes of interacting subpopulations SIR systems Predator-prey systems Populations competing for the same resources Behavior of individuals making up a population Deterministic vs. stochastic Homogeneous vs. spatial

Dynamics of a Single Population Uncounted growth functions; most grow ~exponentially and then taper off All functions are very simple; how can they possibly fit something as complicated as the human population? Possible answer*: Growth is dominated by a few processes at the “right” time scale Faster processes (e.g., signaling, biochemistry) are in steady state Slower processes (changes in climate) are essentially constant Do some math: The few processes at the right scale can lead to simple trends ODEs with one or two equations ~correct at a coarse level *Savageau, MA. PNAS 1979

Growth of a Single Population Typical growth functions Exponential growth Logistic growth (Chapter 4):

Growth of a Single Population Many growth functions may be interpreted as exponential growth with time- (or population-size) dependent growth rates Exponential growth Logistic growth: “growth rate is modulated by distance from K” Gompertz growth: “growth rate is modulated by ratio K/N G ” Richards growth: “growth rate is modulated by function of N R ”

Growth of a Single Population More growth functions may be generated as generalizations of the logistic function Logistic growth: Generalized growth function: “One-variable S-system” Savageau, MA. Math. Biosci. 1980

Extrapolation Growth functions are difficult to extrapolate to final sizes Example: Logistic growth (N1) versus Gompertz growth (N2) PLAS Code N1' = r1 N1 - r1/K N1^2 N2'= R N2 R' = - r2 R r1 =.325 K = 1000 r2 =.1 N1 = 2.2 N2 = 0.1 R = 1 t0 = 0 tf = 20 hr =.1 growth rate decreases exponentially

Very important and very difficult to do Pertinent Example: World Population Source: see Book Fig. 3, Chapter 10 Extrapolation

Reality is More Complicated Conditions change (habitat reduction; climate; …) Stochastic effects New predators; competitors, invasive species, new/fewer food sources; diseases … Example: Commercial fishing (1)Fish a certain percentage (2)Fish at a constant rate

Reality is More Complicated Conditions change (habitat reduction; climate; …) Stochastic effects New predators; competitors, invasive species, new food sources; diseases … Example: Commercial fishing; additional perturbation at time 2.5 (e.g., disease) (1)Fish a certain percentage (2)Fish at a constant rate Time Population Size N1N1 N2N Time N1N1 N2N2

Reality is More Complicated Populations do not have to be macroscopic Cell populations Normal physiology Erythropoiesis Replacement of skin and gut cells Cancer Organismal growth and differentiation Molecular populations

Dynamics of Subpopulations SIR models: groups are human subpopulations Important case: age stratification Recall Leslie model with constant parameters Leslie matrix For four age classes P i : proliferation rates  survival rates  Time- or age-dependent parameters make the model more realistic

Interacting Populations Start with logistic models for two independent populations: Add interactions: Minus sign: inhibitive effect Gause’s exclusion principle (Gause, G.F., The Struggle for Existence, 1934): Two species with exactly the same needs cannot exist in the same constant environment. Usually the needs are slightly different. Questions: Is coexistence possible? Will one species go extinct? Do the two species always coexist? What happens if one goes extinct?

Interacting Populations Numerical Example: Alternative representation: Phase-plane plot

Assess Complete Repertoire of Scenarios As usually, start with steady state: Recognize that trivial solutions exist; divide by N 1 and N 2, respectively: Both “nullclines” are straight lines of N 2 versus N 1.

Assess Entire Repertoire of Scenarios 0K 2 /b K 1 /a Population Size N 2 Population Size N 1 K1K1 N 1 = 0. N 2 = 0. Four steady states! Nullclines: 0 K2K2 N 1 = 0. N 2 = 0.

Assess Entire Repertoire of Scenarios Steady-state lines (“nullclines”) dissect the space in different model responses: K 2 /b K 1 /a Population Size N 2 Population Size N 1 N 1 < 0. N 2 < 0. N 1 = 0. N 2 = 0. N 1 < 0. N 2 > 0. N 1 > 0. N 2 < 0. N 1 > 0. N 2 > K1K1 K2K2

Assess Entire Repertoire of Scenarios Dissection allows us to predict dynamics (and stability of steady states): 0K 2 /b K 1 /a Population Size N 2 Population Size N 1 K1K1 N 1 < 0. N 2 < 0. N 1 = 0. N 2 = 0. N 1 < 0. N 2 > 0. N 1 > 0. N 2 < 0. N 1 > 0. N 2 > 0. 0 K2K2

Assess Entire Repertoire of Scenarios Dynamics exactly on the nullclines: 0K 2 /b K 1 /a Population Size N 2 Population Size N 1 K1K1 N 1 < 0. N 2 < 0. N 1 = 0. N 2 = 0. N 1 < 0. N 2 > 0. N 1 > 0. N 2 < 0. N 1 > 0. N 2 > 0. 0 K2K2

Assess Entire Repertoire of Scenarios Stitch together dynamics for the entire space (qualitatively): 0K 2 /b K 1 /a Population Size N 1 K1K1 0 Population Size N 2 K2K2 N 1 < 0. N 2 < 0. N 1 < 0. N 2 > 0. N 1 > 0. N 2 < 0. N 1 > 0. N 2 > 0.

Assess Entire Repertoire of Scenarios Coexistence depends on location of nullclines (parameter values): 0K 2 /b K 1 /a Population Size N 2 Population Size N 1 K1K1 0 K2K2

Previous Example: Actual Simulations K 2 /b K 1 /a Population Size N 2 Population Size N K1K1 K2K2

Representation of a “Vector Field” Population Size N 2 Population Size N 1

Comparison Population Size N 2 Population Size N 1

Note on Stochasticity Coexistence, existence, and other features of populations can be very strongly affected by stochastic events (e.g., perturbations in the environment). These effects are particularly influential for small population sizes. Why? This is a great concern for the survival of endangered species.

Larger Systems Phase-plane analysis is not very helpful Model format often Lotka-Volterra models (Chapter 4): Recall: strict format, lots of complicated dynamics possible nevertheless!

Applications Traditional examples: Predator-prey systems Competing species Modern examples at the forefront: Microbiomes Metapopulations (see Book, pages )

Spatial Considerations Example: Solenopsis invicta /04/fire-ant-spread-map-pestcemetery.jpg

Spatial Considerations Example: Solenopsis invicta

Spatial Considerations Modeling Approaches: PDEs Simulation “Agent-based models” (ABMs; Book pages ) Each “individual” is an agent Agents are governed by rules of what they can do In addition, there are rules for interactions between agents Simulations are stochastic and very flexible Simulations can yield very complicated results

Easy Implementation: Netlogo Go to Netlogo (>> Download) >> Models Library >> Biology >> Wolf Sheep Predation Set up with defaults Emergency Break: Tools > Halt

Behind the Netlogo Interface

Behind the Netlogo Interface

Exactly the same settings as before

Same settings as before, but account for dynamics of grass

Summary Population dynamics is of great academic and societal interest Base models very simple (functions or simple ODEs) Study: Overall growth (or decline) Carrying capacity Subpopulations Age classes Different types of individuals Spatial expansion Stochastic effects Agent-based models Netlogo