Predator-Prey Population Dynamics Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November 13, 2018 Introduction to Random Processes Predator-Prey Population Dynamics 1
Predator-Prey model (Lotka-Volterra system) Predator-Prey model (Lotka-Volterra system) Stochastic model as continuous-time Markov chain Introduction to Random Processes Predator-Prey Population Dynamics 2
A simple Predator-Prey model ◮ Populations of X prey molecules and Y predator molecules ◮ Three possible reactions (events) → 2 X 1) Prey reproduction: X 2) Prey consumption to generate predator: X + Y → 2 Y Y → ∅ 3) Predator death: ◮ Each prey reproduces at rate α ⇒ Population of X preys ⇒ α X = rate of first reaction ◮ Prey individual consumed by predator individual on chance encounter ⇒ β = Rate of encounters between prey and predator individuals ⇒ X preys and Y predators ⇒ β XY = rate of second reaction ◮ Each predator dies off at rate γ ⇒ Population of Y predators ⇒ γ Y = rate of third reaction Introduction to Random Processes Predator-Prey Population Dynamics 3
The Lotka-Volterra equations ◮ Study population dynamics ⇒ X ( t ) and Y ( t ) as functions of time t ◮ Conventional approach: model via system of differential eqs. ⇒ Lotka-Volterra (LV) system of differential equations ◮ Change in prey ( dX ( t ) / dt ) = Prey generation - Prey consumption ⇒ Prey is generated when it reproduces (rate α X ( t )) ⇒ Prey consumed by predators (rate β X ( t ) Y ( t )) dX ( t ) = α X ( t ) − β X ( t ) Y ( t ) dt ◮ Predator change ( dY ( t ) / dt ) = Predator generation - consumption ⇒ Predator is generated when it consumes prey (rate β X ( t ) Y ( t )) ⇒ Predator consumed when it dies off (rate γ Y ( t )) dY ( t ) = β X ( t ) Y ( t ) − γ Y ( t ) dt Introduction to Random Processes Predator-Prey Population Dynamics 4
Solution of the Lotka-Volterra equations ◮ LV equations are non-linear but can be solved numerically 18 X (Prey) ◮ Prey reproduction rate α = 1 Y (Predator) 16 14 ◮ Predator death rate γ = 0 . 1 12 Population Size ◮ Predator consumption of prey β = 0 . 1 10 8 ◮ Initial state X (0) = 4, Y (0) = 10 6 4 ◮ Boom and bust cycles 2 0 0 10 20 30 40 50 60 70 80 90 100 Time ◮ Start with prey reproduction > consumption ⇒ prey X ( t ) increases ◮ Predator production picks up (proportional to X ( t ) Y ( t )) ◮ Predator production > death ⇒ predator Y ( t ) increases ◮ Eventually prey reproduction < consumption ⇒ prey X ( t ) decreases ◮ Predator production slows down (proportional to X ( t ) Y ( t )) ◮ Predator production < death ⇒ predator Y ( t ) decreases ◮ Prey reproduction > consumption (start over) Introduction to Random Processes Predator-Prey Population Dynamics 5
State-space diagram ◮ State-space diagram ⇒ plot Y ( t ) versus X ( t ) ⇒ Constrained to single orbit given by initial state ( X (0) , Y (0)) 40 35 30 25 (X(0),Y(0)) = Y (Predator) (4,1) 20 (4,2) 15 (4,4) (4,6) 10 (4,10) 5 0 0 2 4 6 8 10 12 14 16 18 20 X (Prey) Buildup: Prey increases fast, predator increases slowly (move right and slightly up) Boom: Predator increases fast depleting prey (move up and left) Bust: When prey is depleted predator collapses (move down almost straight) Introduction to Random Processes Predator-Prey Population Dynamics 6
Two observations ◮ Too much regularity for a natural system (exact periodicity forever) 20 X (Prey) Y (Predator) 15 Population Size 10 5 0 0 100 200 300 400 500 600 700 800 900 1000 Time ◮ X ( t ), Y ( t ) modeled as continuous but actually discrete. Is this a problem? ◮ If X ( t ), Y ( t ) large can interpret as 18 X (Prey) concentrations (molecules/volume) Y (Predator) 16 14 ⇒ Often accurate (millions of molecules) 12 Population Size 10 ◮ If X ( t ), Y ( t ) small does not make sense 8 X≈0.07 6 ⇒ We had 7 / 100 prey at some point! 4 2 ◮ There is an extinction event we are missing 0 0 10 20 30 40 50 60 70 80 90 100 Time Introduction to Random Processes Predator-Prey Population Dynamics 7
Things deterministic model explains (or does not) ◮ Deterministic model is useful ⇒ Boom and bust cycles ⇒ Important property that the model predicts and explains ◮ But it does not capture some aspects of the system ⇒ Non-discrete population sizes (unrealistic fractional molecules) ⇒ No random variation (unrealistic regularity) ◮ Possibly missing important phenomena ⇒ Extinction ◮ Shortcomings most pronounced when number of molecules is small ⇒ Biochemistry at cellular level (1 ∼ 5 molecules typical) ◮ Address these shortcomings through a stochastic model Introduction to Random Processes Predator-Prey Population Dynamics 8
Stochastic model as CTMC Predator-Prey model (Lotka-Volterra system) Stochastic model as continuous-time Markov chain Introduction to Random Processes Predator-Prey Population Dynamics 9
Stochastic model ◮ Three possible reactions (events) occurring at rates c 1 , c 2 and c 3 c 1 1) Prey reproduction: → 2 X X c 2 → 2 Y 2) Prey consumption to generate predator: X + Y c 3 3) Predator death: Y → ∅ ◮ Denote as X ( t ) , Y ( t ) the number of molecules by time t ◮ Can model X ( t ) , Y ( t ) as continuous time Markov chains (CTMCs)? ◮ Large population size argument not applicable ⇒ Interest in systems with small number of molecules/individuals Introduction to Random Processes Predator-Prey Population Dynamics 10
Stochastic model (continued) ◮ Consider system with 1 prey molecule x and 1 predator molecule y ◮ Let T 2 (1 , 1) be the time until x reacts with y ⇒ Time until x , y meet, and x and y move randomly around ⇒ Reasonable to model T 2 (1 , 1) as memoryless � � T 2 (1 , 1) > s � � P T 2 (1 , 1) > s + t = P ( T 2 (1 , 1) > t ) ◮ T 2 (1 , 1) is exponential with parameter (rate) c 2 Introduction to Random Processes Predator-Prey Population Dynamics 11
Stochastic model (continued) ◮ Suppose now there are X preys and Y predators ⇒ There are XY possible predator-prey reactions ◮ Let T 2 ( X , Y ) be the time until the first of these reactions occurs ◮ Min. of exponential RVs is exponential with summed parameters ⇒ T 2 ( X , Y ) is exponential with parameter c 2 XY ◮ Likewise, time until first reaction of type 1 is T 1 ( X ) ∼ exp( c 1 X ) ◮ Time until first reaction of type 3 is T 3 ( Y ) ∼ exp( c 3 Y ) Introduction to Random Processes Predator-Prey Population Dynamics 12
CTMC model ◮ If reaction times are exponential can model as CTMC ⇒ CTMC state ( X , Y ) with nr. of prey and predator molecules c 1 ( X − 1) X − 1 , Y +1 X , Y +1 Transition rates ◮ ( X , Y ) → ( X + 1 , Y ): c 3 ( Y + 1) c 3 ( Y + 1) c 2 ( X + 1) Y c 2 XY Reaction 1 = c 1 X ◮ ( X , Y ) → ( X − 1 , Y +1): c 1 ( X − 1) c 1 X Reaction 2 = c 2 XY X − 1 , Y X +1 , Y X , Y ◮ ( X , Y ) → ( X , Y − 1): c 2 ( X + 1)( Y − 1) c 2 X ( Y − 1) Reaction 3 = c 3 Y c 3 Y c 3 Y ◮ State-dependent rates c 1 X X , Y − 1 X +1 , Y − 1 Introduction to Random Processes Predator-Prey Population Dynamics 13
Simulation of CTMC model ◮ Use CTMC model to simulate predator-prey dynamics ◮ Initial conditions are X (0) = 50 preys and Y (0) = 100 predators 400 X (Prey) Y (Predator) 350 ◮ Prey reproduction rate 300 c 1 = 1 reactions/second 250 Population Size ◮ Rate of predator consumption of prey 200 c 2 = 0 . 005 reactions/second 150 ◮ Predator death rate 100 c 3 = 0 . 6 reactions/second 50 0 0 5 10 15 20 25 30 Time ◮ Boom and bust cycles still the dominant feature of the system ⇒ But random fluctuations are apparent Introduction to Random Processes Predator-Prey Population Dynamics 14
CTMC model in state space ◮ Plot Y ( t ) versus X ( t ) for the CTMC ⇒ state-space representation 400 350 300 Y (Predator) 250 200 150 100 50 0 50 100 150 200 250 300 350 X (Prey) ◮ No single fixed orbit as before ⇒ Randomly perturbed version of deterministic orbit Introduction to Random Processes Predator-Prey Population Dynamics 15
Effect of different initial population sizes ◮ Chance of extinction captured by CTMC model (top plots) X(0) = 8, Y(0) = 16 X(0) = 16, Y(0) = 32 X(0) = 32, Y(0) = 64 2000 1000 700 X (Prey) X (Prey) X (Prey) Y (Predator) Y (Predator) 600 Y (Predator) 800 1500 500 Population Size Population Size Population Size 600 400 1000 300 400 200 500 200 100 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 Time Time Time X(0) = 64, Y(0) = 128 X(0) = 128, Y(0) = 256 X(0) = 256, Y(0) = 512 500 350 1000 X (Prey) X (Prey) X (Prey) Y (Predator) 300 Y (Predator) Y (Predator) 400 800 250 Population Size Population Size Population Size 300 600 200 150 200 400 100 100 200 50 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 Time Time Time (Notice that Y-axis scales are different) Introduction to Random Processes Predator-Prey Population Dynamics 16
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