1. Recap from last week
Models are useful as thinking aids, not just for quantitative prediction
Building models helps us to crystallize our questions and simplify conceptually (analogy to microcosm experiments)
complex models aren’t necessarily useful or accurate!
Flowchart approach to building models:
boxes are state variables,
arrows indicate processes that increase or decrease state variables: assigning functional forms to these arrows is how we specify mechanism
Differential equations represent the net effect of these arrows on the rate of change of our state variable

2. Quiz results
1 strongly agree, 2 agree, 3 neutral, 4 disagree, 5 strongly disagree
I understand how and why mathematical models are used in Ecology 1.9
I feel comfortable critically reading or reviewing ecology papers featuring mathematical models 3.5
I know how to build models to answer ecological questions 3.7
I can perform some simple analysis of math models (e.g. curve sketching or ”back of the envelope” calculations 3.5
I plan to incorporate some mathematical modeling in my research 1.9

3. Logistic growth revisited
“Woolly” question – what drives/regulates population growth?
Malthus and Verhulst proposed that resources (food) limit growth
This equation seems to do a nice job of describing growth of some populations, but why? Where did it come from?
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾

4. Resource limitation and population size

5. Resource limitation and population size

6. Resource limitation and population size

7. Resource limitation and population size

8. Resource limitation and population size

9. Aside from being hangry….
… what does this mean for individual fitness?

10. Boxes and arrows revisited
Population changes through births and deaths N
births
deaths

11. Boxes and arrows revisited
Population changes through births and deaths N
births
deaths
Total birth rate = per capita birth rate x population size
Total death rate = per capita death rate x population size

12. Mechanism
How does the population size influence per capita birth and death rate? (1) Sketch it!
per capita birth rate
per capita death rate
per capita birth rate
population size (N)
population size (N)

13. Mechanism
How does the population size influence per capita birth and death rate? (1) Sketch it!
per capita birth rate
per capita death rate
per capita birth rate
population size
population size
More people = less pie = higher starvation risk
More people = less pie = less babies

14. Mechanism
How does the population size influence per capita birth and death rate? (2) Write down the simplest functional form for your relationship
per capita birth rate
per capita death rate
per capita birth rate
population size (N)
population size (N)
per capita death rate = d0 + d1N
per capita birth rate = b0 – b1N

15. Translate into equation
In words: Rate of change of population size = total birth rate – total death rate dN/dt = (b0 – b1N) x N – (d0 + d1N) x N
per capita birth rate
per capita death rate
per capita birth rate
population size
population size
per capita birth rate = b0 – b1N
per capita death rate = d0 + d1N

16. Over to you:
How do I get from this… dN/dt = (b0 – b1N) x N – (d0 + d1N) x N … to this? 𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 In other words, how are r and K related to b0, b1, d0 and d1? What does this tell us about their meanings?

17. Logistic growth - summary
𝑟= 𝑏 0 − 𝑑 0 𝐾= 𝑏 0 − 𝑑 0 𝑏 1 + 𝑑 1 r is the maximum population growth rate Increasing density dependence in the birth or death rate (or both) reduces K (carrying capacity) b0 d0

18. Hunting revisited
So, my wild species exhibits logistic growth via resource limitation, but what else might limit its growth? Hunting by predators or people. Suppose we manage a fishing fleet where we decide how many vessels to deploy Because we have Ecology degrees, we’re interested in sustainable fishing as well as making money

19. Questions
For a given amount of hunting effort (e.g. boats deployed), what happens to my population size, and my fishing yield?

20. N Conceptual model How would you modify this to account for hunting?
births
deaths

21. Conceptual model N
births
deaths
Removal by hunting

22. Mechanism
Let’s call our hunting effort E. How do we think E and N influence the hunting rate (no of animals removed per unit time?) N
births
deaths
Removal by hunting

23. Deriving a functional form for hunting

24. Deriving a functional form for hunting

25. Deriving a functional form for hunting

26. Deriving a functional form for hunting

27. Deriving a functional form for hunting
Doubling effort doubles harvest

28. Deriving a functional form for hunting

29. Deriving a functional form for hunting
Doubling population doubles harvest

30. Deriving a functional form for hunting
So, a reasonable first guess for harvest rate (catch per unit time)
is q x E x N (what is q?)

31. Building the model
In words: rate of change of population = growth rate (logistic) – harvest rate
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 −𝑞𝐸𝑁

32. Over to you
Our “wooly” question: for a given amount of hunting effort (e.g. boats deployed), what happens to my population size, and my fishing yield?
Make the question less wooly:
specify the time scale. Are we talking about long time scales? Does deploying a constant hunting effort result in a steady state or equilibrium?
What are the response variables?
Population size = N evaluated at equilibrium
Yield = amount harvested per unit time = qEN

33. Over to you
Solve this model at equilibrium, and sketch population size (N) and yield (qEN) as functions of hunting effort (E)

34. Open access hunting
Unfortunately, hunting isn’t usually managed by a sustainability- minded monopoly
Hunting of many wild species is unregulated, poorly enforced or poached
Question of conservation concern – under open access hunting, how does hunting effort change through time?

35. Hunting effort
Let’s measure hunting effort as number of hunters Hunting effort increases when more hunters are attracted to the profession, and decreases when they give up hunting – but what determines these rates? E

36. Hunting effort
Let’s measure hunting effort as number of hunters Hunting effort increases when more hunters are attracted to the profession, and decreases when they give up hunting – but what determines these rates? PROFIT E
- $$
+ $$$$$

37. Mechanism
Suppose the market price per unit catch is p One unit of hunting effort (E=1) yields a catch of qN per unit time Suppose the cost of a unit hunting effort = c Hunters will be attracted pqN >c, and deterred if pqN <c E
- $$
+ $$$$$

38. Equation
Rate of change of total hunting effort is proportional to total hunting effort x individual profit So are we done?
𝑑𝐸 𝑑𝑡 =𝛼𝐸 𝑝𝑞𝑁−𝑐 E

39. Coupling the human and natural systems
Population decreases as exploitation increases, which reduces profit and slows increase in hunting effort – need to couple dynamics of the population to that of exploitation effort N
‘natural’ deaths
births
𝑑𝑁 𝑑𝑡 =𝑟𝑁 1− 𝑁 𝐾 −𝑞𝐸𝑁
‘hunting’ deaths E
𝑑𝐸 𝑑𝑡 =𝛼𝐸 𝑝𝑞𝑁−𝑐
Total income
(price x catch)
Total expenditure
(cost x effort)