FW364 Ecological Problem Solving Class 20: Predation November 12, 2013

Outline for Today Continuing with our shift in focus from populations of single species to community interactions Objectives from Last Class: Introduced coupled predator-prey interactions Introduced basic predator-prey equations Objectives for Today: Application of calculus to predator-prey dynamics Continue building predator predator-prey equations Look at a very important example of predator-prey interactions Return the mid-term 2 Objectives for Next Two Classes: Link predator-prey equations Develop predator-prey models using different assumptions No textbook chapters!

Recap from Last Class Vt+1 = Vt + Bv - Dv Pt+1 = Pt + Bp - Dp Developed separate (not explicitly linked) predator and prey equations Started with stock and flow models: Prey Births Deaths Predator Births Deaths Revised familiar equation: Nt+1 = Nt + B - D Victim, V: Vt+1 = Vt + Bv - Dv Predator, P: Pt+1 = Pt + Bp - Dp

Which in a natural system look like: Recap from Last Class Victim, V: Vt+1 = Vt + Bv - Dv Predator, P: Pt+1 = Pt + Bp - Dp Dynamics go like this: Prey go up… Predators go up… Prey go down… Predators go down…  predators go up  prey go down  predators go down  prey go up Which in a natural system look like:

Next Step Vt+1 = Vt + Bv - Dv Pt+1 = Pt + Bp - Dp Victim, V: Predator, P: Pt+1 = Pt + Bp - Dp Next Step: Link (explicitly) these two equations (Wednesday’s class) We need shared variables/parameters in both equations i.e., Some predator variables/parameters need to be in the prey equation Some prey variables/parameters need to be in the predator equation

Next Step Vt+1 = Vt + Bv - Dv Pt+1 = Pt + Bp - Dp Victim, V: Predator, P: Pt+1 = Pt + Bp - Dp Next Step: BUT FIRST To link equations, we’ll need to investigate dynamics of predators and prey at the same time, so we need to switch from discrete time to continuous time i.e., we will use instantaneous rates to study predator-prey interactions (and other two species interactions)

Recall difference between discrete and continuous growth: Remember This? Recall difference between discrete and continuous growth: Discrete Continuous Population Size Population Size Nt+1 = Nt λ dN/dt = r N Nt = N0 λt Nt = N0 ert

Let’s consider a driving example: Calculus Refresher Let’s consider a driving example: Imagine we drive to MBC in Webberville Say it takes us ½ hour to go 20 miles What was our average speed (in mi/hr)?  20 miles / ½ hr = 40 mi/hr So, 40 mi/hr is our average speed, but we were not going 40 mi/hr the whole trip i.e., our instantaneous speed (i.e., speed on speedometer) varied  Started slower than 40mi/hr in East Lansing  Drove faster than 40mi/hr on I-96  Drove slower than 40mi/hr when got off interstate in Webberville

Calculus Refresher Let’s look at a figure Let’s consider a driving example: Imagine we drive to MBC in Webberville Say it takes us ½ hour to go 20 miles What was our average speed (in mi/hr)?  20 miles / ½ hr = 40 mi/hr Average speed ~ finite rate Like our discrete time rates (b’, d’, λ) (average rate calculated between two end points) Speed on speedometer ~ instantaneous rate Like our continuous time rates (b, d, r) (constantly changing rate) Let’s look at a figure

Let’s consider a driving example: Calculus Refresher Let’s consider a driving example:

Let’s consider a driving example: Calculus Refresher Let’s consider a driving example: Slope of the curve is equivalent to our speed Slope = Rise Run ΔY ΔX = Speed = Distance Time

Let’s consider a driving example: Calculus Refresher Let’s consider a driving example: End slow: Shallow slope Fast on I-96: Steep slope Start out slow: Shallow slope

Let’s consider a driving example: Calculus Refresher Let’s consider a driving example: Average speed for trip: Slope of line connecting end points Slope = 20 mi 30 min 20 mi 0.5 hr = = 40 mi/hr Finite rates are average rates of change over some definite interval e.g., between two locations; between two years

Calculus Refresher We can determine our speed over shorter intervals by calculating the slope of a tangent to the curve at any point

Calculus Refresher To obtain our exact speed at any given moment in time (i.e., the speed on our speedometer at any moment), we need a method to determine the slope of the curve at each moment If we define a moment as an infinitesimally small interval of time, we have a definition of our instantaneous rate of speed  slope of function over infinitely small interval Trick to finding the slope: calculus! We can find the instantaneous rate (which in our example is not a single number b/c the slope constantly changes) by differentiating the distance vs. time function (i.e., taking the derivative) and plugging in a value for time

Calculus Refresher To obtain our exact speed at any given moment in time (i.e., the speed on our speedometer at any moment), we need a method to determine the slope of the curve at each moment If we define a moment as an infinitesimally small interval of time, we have a definition of our instantaneous rate of speed  slope of function over infinitely small interval Trick to finding the slope: calculus! We can find the instantaneous rate (which in our example is not a single number b/c the slope constantly changes) by differentiating the distance vs. time function (i.e., taking the derivative) and plugging in a value for time We’ll use the tangent method to approximate (for exercises), and software (Stella) to do calculus for labs We are not doing the differentiation in this class

Calculus Refresher “Distance vs. time is great for a physics major, but what does this have to do with ecology and management?”

Calculus Refresher “Distance vs. time is great for a physics major, but what does this have to do with ecology and management?” Curve could describe dingo population growth after introduction to a new area Abundance, N (years)

Calculus Refresher “Distance vs. time is great for a physics major, but what does this have to do with ecology and management?” To determine the instantaneous net growth rate for the population, we would need to define an equation: N = f(time)  Taking the derivative of that function, dN/dt (remember: dN/dt is like ΔN/Δt), would give us the instantaneous net growth rate of population Abundance, N (years)

Population Growth Example Let’s do a population growth exercise with Daphnia

Population Growth Example Q1: What is the average population growth rate for the 30 day period?

Population Growth Example Q1: What is the average population growth rate for the 30 day period? 900 – 600 Daphnia / m3 Average = = 10 Daphnia / m3 / day 30 days

Population Growth Example Q2: What is the growth rate at points A and B (using tangent method)? B A

Population Growth Example Q2: What is the growth rate at points A and B (using tangent method)? B A 1000 – 800 Daphnia / m3 Slope A = = 20 Daphnia / m3 / day 15 - 5 day

Population Growth Example Q2: What is the growth rate at points A and B (using tangent method)? B A 920 – 1020 Daphnia / m3 Slope B = = -10 Daphnia / m3 / day 30 - 20 days

Population Growth Example Q3: For what days was the (instantaneous) growth rate: a) negative, b) positive, and c) zero?

Population Growth Example Q3: For what days was the (instantaneous) growth rate: a) negative, b) positive, and c) zero? dN/dt Positive dN/dt Negative dN/dt Zero

Instantaneous Rates & Predator-Prey Models We’ll need to use instantaneous rates to analyze predator-prey dynamics! Why? We are interested in how predators and prey interact  Interactions will happen in continuous time i.e., want to investigate linked changes in prey abundance over time (ΔV / Δt or dV / dt) and changes in predator abundance over time (ΔP / Δt or dP / dt) Let’s continue to build our predator prey models 

Instantaneous Rates & Predator-Prey Models Victim, V: Vt+1 = Vt + Bv - Dv Predator, P: Pt+1 = Pt + Bp - Dp Start with the simple models we built last class Well do prey first, then predators

Instantaneous Rates & Predator-Prey Models Victim, V: Vt+1 = Vt + Bv - Dv Predator, P: Pt+1 = Pt + Bp - Dp Rearrange to: Vt+1 - Vt = Bv - Dv And substitute ΔV for Vt+1 - Vt ΔV = Bv - Dv Still a model of discrete change in V To make continuous, change time to “infinitely small time intervals” (~instantaneous time step) dV/dt = Bv - Dv

Instantaneous Rates & Predator-Prey Models Victim, V: Vt+1 = Vt + Bv - Dv Predator, P: Pt+1 = Pt + Bp - Dp Rearrange to: Rearrange to: Vt+1 - Vt = Bv - Dv Pt+1 - Pt = Bp - Dp And substitute ΔV for Vt+1 - Vt And substitute ΔP for Pt+1 - Pt ΔV = Bv - Dv ΔP = Bp - Dp Still a model of discrete change in V Still a model of discrete change in P To make continuous, change time to “infinitely small time intervals” (~instantaneous time step) To make continuous, change time to “infinitely small time intervals” (~instantaneous time step) dV/dt = Bv - Dv dP/dt = Bp - Dp

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = Bv - Dv Predator, P: dP/dt = Bp - Dp A few comments on these equations: When changing from discrete to continuous time, we added time units: dV/dt, dP/dt, Bv, Bp, Dv, and Dp units are: Numbers/time dV/dt and dP/dt can be positive or negative, depending on whether population is increasing or decreasing Next step: Express B and D as products of per capita birth (b) and death (d) rates and population sizes (P and V)

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = Bv - Dv Predator, P: dP/dt = Bp - Dp Birth rate: Bv = bvV where bv is the per capita birth rate Units: bv = # prey (babies) 1 prey (adult) * time = 1 time bvV = prey time i.e., # prey born in pop per unit time

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = Bv - Dv Predator, P: dP/dt = Bp - Dp Birth rate: Bv = bvV where bv is the per capita birth rate Birth rate: Bp = bpP where bp is the per capita birth rate Units: bv = # prey (babies) 1 prey (adult) * time = 1 time Units: bp = # predator (babies) 1 pred. (adult) * time = 1 time bvV = prey time bpP = predators time i.e., # prey born in pop per unit time i.e., # preds born in pop per unit time

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = bvV - Dv Predator, P: dP/dt = bpP - Dp

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = bvV - Dv Predator, P: dP/dt = bpP - Dp Death rate: Dv = dvV where dv is the per capita death rate Units: dv = [Prob. individual dies] time = 1 dvV = prey time i.e., # prey that die in pop per unit time

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = bvV - Dv Predator, P: dP/dt = bpP - Dp Death rate: Dv = dvV where dv is the per capita death rate Death rate: Dp = dpP where dp is the per capita death rate Units: dv = [Prob. individual dies] time = 1 Units: dp = [Prob. individual dies] time = 1 dvV = prey time i.e., # prey that die in pop per unit time dpP = predators time i.e., # preds that die in pop per unit time

Instantaneous Rates & Predator-Prey Models Victim, V: dV/dt = bvV - dvV Predator, P: dP/dt = bpP - dpP Look familiar? These equations are the same as what we did earlier when we were studying individual populations with continuous growth i.e., dN/dt = bN - dN A few notes: Remember: b and d do not have the primes ( ’ ) because continuous time Equations describe rates of change in predator and prey populations Next step: Link equations

Link predator-prey equations Handing back exams in Lab Today Looking Ahead No class tomomrrow! Next Class (Monday): Link predator-prey equations Discuss assumptions Homework Isle Royale Wolf Paper Handing back exams in Lab Today