Models

Question 1

How is a population model generally described?

Answer 1

R(t): growth rate of population over one year

Question 2

How can constant population growth be described?

What is its solution?

Answer 2

Growth rate of the population over one year R(t) is constant: R(t) = ρ

ρ: constant growth rate

Question 3

What is the Malthusian population model?

How can it be described?

What is its solution?

Answer 3

Exponential population model.

Growth rate of the population over one year R(t) linearly grows with N(t): R(t) = r * N(t)

with r: intrinsic growth rate

Question 4

What is Verhulst’s population model?

How can it be described?

What do de parameters represent?

Answer 4

Logistic population model with intraspesific competition (limited resources).

Two possible parametrisations for the growth rate of the population over one year:

• “r-K” parametrisation: R(t) = r * N * (1- N/K)
• “r-α” parametrisation: R(t) = r * N - α * N²

with:
r: intrinsic growth rate,
α: coefficient of intraspecific competition,
K: carrying capacity parameter (lim N(t) = K)

Question 5

Whats the solution of Verhulst’s model in the “r-K” parametrisation?

How is K represented in a graph of N(t)?

Answer 5

K is the threshold of the population size.

with:
r: intrinsic growth rate,
K: carrying capacity parameter (lim N(t) = K)

Question 6

What are examples for population models?

Answer 6

Malthusian (or exponential) population model

Verhulst’s (or logistic) population model

Question 7

What is Kermack-McKendrick’s epidemic model?

How can it be described?

Answer 7

Structured population model

SIR model: Susceptible, Infected & Recovered

with:
S(t): number of susceptible, I(t): number of infected
β: infection rate, α: recovery or death rate

Question 8

What is Bernoulli’s smallpox model?

Answer 8

Aim to compute life expectancy without smallpox

Is inoculatin beneficial for humanity?

Goal to describe the relationship between the number of survivors with and without smallpox

information/assumptions:

yearly smallpox infection rate (fraction of susceptible people who get infected per year): β = 1/8,
probability of dying from the smallpox once infected: ν = 1/8,
probability of dying from inoculation is 1/200

Question 9

How can life expectancy be described in general?

Answer 9

with
P(t): number of individuals still alive at beginning of year t > 0, i.e. the so called number of survivors at year t,
P(0): number of newborns

Question 10

How can the number of survivors without smallpox be described?

Answer 10

with
P(t): number of individuals still alive at beginning of year t > 0, i.e. the so called number of survivors at year t,
β: infection rate, ν: probability of dying from smallpox

Question 11

How can the life expectancy without smallpox be described?

Answer 11

with:
Q(t): number of survivors without smallpox

Question 12

What is the Lotka-Volterra model?

What different vartiations are there?

Answer 12

Interacting populations model with prey and predator

Three main variations:

• Original
• Logistic growth for prey (or predator)
• Functional response (how prey and predator interact)
• Generalised

Question 13

What is the original Lotka-Volterra model?

How can it be described?

Answer 13

Prey population in absence of predator assumed to increase exponentially and in absence of prey, predator population decreases exponentially

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ

Question 14

How can the Lotka-Volterra model with logistic grwoth for the prey be described?

Answer 14

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ,
α: intraspecific competition

Question 15

What is the Lotka-Volterra model with functional response?

How can it be described?

Answer 15

Describes how prey and predator interact, by considering the behaviour of a single predator in the system.

with
f(N): functional response, i.e. the number of prey eaten per predator and per unit of time,
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
ε: conversion efficiency

Question 16

What is the Rosenzweig-MacArthur model?

How can it be desrcibed?

Answer 16

Functional response that is widely used in theoretical ecology and often with the Lotka-Volterra model with logistic growth of the prey. It is a Holling type II functional response.

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, ε: conversion efficiency,
α: intraspecific competition,
T_h: handling time (time required to handle a single prey)

Question 17

What is the generalised Lotka-Volterra model?

How can it be described?

Answer 17

Original Lotka-Volterra model written in matrix notation.

Easy to expand number of species as a set of equations for S species.

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ,
α: Matrix with intraspecific and interspecific competition

Question 18

What is the Luria-Debrück model?

Answer 18

Describes resistance of bacteria against bacteriophages.

Two statistical models:

• Contact Hypothesis (Hc): bacterial resistance acquired by contact with virus
• Mutation Hypothesis (Hμ): bacterial resistance acquired by spontaneous mutation

Both transmit resistance to next generation.

Experimental results clearly support Hμ

Number of bacteria at generation time t: N(t) = N(0) * 2^t

Question 19

What are the equilibrium points of Verhulst’s model?

How can their stability be determined?

How does K relate to the equilibrium point?

Answer 19

x-Axis intersections in a Graph of time derivative of population size dn/dt over Population size (N)

Local stability analysis: See if the population returns to the equilibrium point when you move up of down in population size

–> stable: moves back to equilibrium point

–> unstable: moves further away

K is the maximum in population size that is an equilibrium point

Question 20

How does a trajectory analysis of Verhulst’s model work?

What is the inflection point?

Answer 20

Inverse Graph and draw figure next to it with population size N over time.

Value of horizontal axis of initial graph gives speed at which population size changes.

The inflection point is the maximum of N (second derivative of N(t) equals 0)

Question 21

What are the equilibrium points of the Kermack-McKendrik (SI) model?

What are the signs of the derivatives?

What is the epidemic threshold?

Answer 21

• *Equilibrium points**: Time derivative of both DE are 0.
• -> Set of equilibrium points: I* = 0 and S* >= 0

Signs:

dS/dt < 0 if S > 0 and I > 0, which is always the case

Epidemic threshold: Epidemic peak reached for: S(t) = α/β

with:
S(t): number of susceptible, I(t): number of infected
β: infection rate, α: recovery or death rate

Question 22

What is the first integral in the Kermack-McKendrik model?

For what can it be used?

Answer 22

First integral: V(S, I)

• *Note**: V(S(t), I(t)) = V(S(0), I(0))
• -> first integral remains constant for any time t/along the trajectory
• -> Better to take a higher t than 0, because if there are more infected, the error will most likely be smaller

With the above equation we can solve for the epidemic threshold and so by using the fact that α = 1/(Ø days of infection) we can calculate α and β.

And we can determine the value of I(t) at the epidemic threshold by inserting S(t) = α/β into the equation and solving for I(t).

with:
S(t): number of susceptible, I(t): number of infected
β: infection rate, α: recovery or death rate

Question 23

What are the non-trivial zero growth isoclines for the original Lotka-Volterra model?

What are the equilibrium points?

Answer 23

• *Non-trivial zero growth isoclines**: separate regions with different signs of the derivatives (regions of phase space). They can be determined by setting the DE to 0 and solving for the other species.
• *Prey**: P = r/γ, Predator: N = m/β
• *Equilibrium points:**
• *Trivial**: N* = 0, P * = 0
• *Non-trivial**: intersection of non-trivial zero growth isoclines

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ

Question 24

What are the non-trivial zero growth isoclines for the Lotka-Volterra model with logistic growth for the prey?

What are the equilibrium points?

Answer 24

• *Non-trivial zero growth isoclines:**
• *Predator**: N = m/β, Prey: P = r/γ - (α/γ) * N

Equilibrium points: Solve for N* and then calculate P*
N* = m/β and P* = r/γ - (α/γ) * (m/β) = r/γ * (1 - m/(K*β))

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ,
α: intraspecific competition,
K: carrying capacity parameter (lim N(t) = K)

Question 25

What is the first integral of the original Lotka-Volterra model?

What does it tell us about the trajectories and the equilibrium point?

Answer 25

Again, V(N,P) is constant along trajectories (i.e. dV/dt = 0)

If we add or remove a small number of prey and/or predator we will not go back toward the equilibrium point, nor diverge toward oscillations of increasing amplitude

–> neutral or marginal stability

with
N(t): prey population, P(t): predator population,
r: intrinsic growth rate of N, m: mortality rate of P,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ

Question 26

What is the first integral of the Lotka-Volterra model with logistic growth for the prey?

What does it tell us about the trajectories and the equilibrium point?

Answer 26

No longer constant along the trajectories, but decreases until the trajectories converge to the equilibrium point.

V(P, N) is now called a Lyapunov function.

with
N(t): prey population, P(t): predator population,
N*: equilibrium point
β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ,
α: intraspecific competition

Question 27

What is the first integral of the Lotka-Volterra model in general?

Answer 27

Solve by replacing 1/N * dN/dt and 1/P * dP/dt with model equations

with
N(t): prey population, P(t): predator population,
N* and P*: equilibrium points,
γ: attack rate, β: attack rate and conversion (β = γ * ε),
ε: conversion efficiency, in general: β < γ

Question 28

How does the phase space of the Kermack-McKendrick (SI) model look?

Answer 28

6 “regions”:

1. The light blue region (I > 0 and S > α/β) where the derivative for I is positive (they increase) and that for S is negative (they decrease)
2. The light yellow region (I > 0 and S < α/β) where I and S decrease
3. The blue zero-growth isocline positioned on the epidemic threshold (I > 0 and S = α/β) where only S decrease
4. The strictly positive part of the X-axis (I = 0 and S > 0) in green, where both derivatives for S and I equal 0 (starting on this axis, the number of susceptible remains constant)
5. The orange Y-axis (I > 0 and S = 0), where the infected will decrease exponentially toward 0
6. The red point given by the origin of both axes, i.e., the trivial equilibrium

Question 29

How does the phase space of the original Lotka-Volterra model look?

Answer 29

12 regions:

1-2) the trivial and non-trivial equilibrium points
3-4) the two axes; 5-8) the zero growth isoclines that are separated by the non-trivial equilibrium point
9-12) the four regions for which the signs of the derivatives are different. The grey vectors give the direction of the trajectories for regions 5 to 12

Question 30

How does the phase space of the Lotka-Volterra model with logistic growth for the prey look like?

Answer 30

12 regions:

1-2) the trivial and non-trivial equilibrium points
3-4) the two axes; 5-8) the zero growth isoclines that are separated by the non-trivial equilibrium point
9-12) the four regions for which the signs of the derivatives are different. The grey vectors give the direction of the trajectories for regions 5 to 12