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

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

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

Resource limitation and population size

Resource limitation and population size

Resource limitation and population size

Resource limitation and population size

Resource limitation and population size

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

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

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

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)

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

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

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

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?

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

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

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

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

Conceptual model N births deaths Removal by hunting

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

Deriving a functional form for hunting

Deriving a functional form for hunting

Deriving a functional form for hunting

Deriving a functional form for hunting

Deriving a functional form for hunting Doubling effort doubles harvest

Deriving a functional form for hunting

Deriving a functional form for hunting Doubling population doubles harvest

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?)

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

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

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

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?

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

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 - $$ + $$$$$

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 - $$ + $$$$$

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

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)