Predators, prey and prevalence Image from by Andrew Bate Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Bath, UK

Timeline Intro to Eco-epidemiology Endemic thresholds in PP oscillations Break Complex dynamics Disease in group-defending prey #Work in progress#

Eco-epidemiology Ecology: dynamics from interactions between different species, e.g. predator—prey Epidemiology: dynamics of disease in host population Eco-epidemiology: dynamics from interactions between different species where one or more are a host of an infectious disease

Examples of eco-epidemiology Grey-squirrel—Red-squirrel—Squirrelpox Myxomatosis’s knock effects on species that interact with rabbits Moose—Wolves— Canine-Parvovirus And many others…

Making an eco-epidemiological model Underlying ecology Underlying epidemiology Underlying interaction of ecology and epidemiology For the moment, we will only consider predator—prey ODE models

Underlying Ecology (no disease) What at the prey dynamics in absence of predators? (logistic, Allee effect) Do predators attack susceptible prey? Is the predator specialist or generalist? What are the predators’ underlying dynamics (functional and numerical responses)?

“Our” predator—prey model NP Image from biosulf.org

Predator—prey: Results Three scenarios: Prey only: prey grow to carrying capacity: stable if: Predator—prey steady state: stable coexistent equilibrium exist if: Predator—prey oscillations: stable coexistent cycles exist if:

Underlying Epidemiology (no ecological interactions) Is infection macro or microparasitic? What stages of infection are there (latency, recovery, immunity)? How is the disease transmitted? What is the force of infection? What are the consequences of infections (ignoring interaction effects)?

“Our” Epidemiology: SI disease Populations split into two distinct classes: Susceptible and Infected, i.e. S(t)+I(t)=N(t). Density dependent force of infection SI InfectionBirths Disease- related deaths Natural deaths

Simplified SI disease Assuming population is constant, we can reduce down to one equation and non- dimensionalise to get: where

Simplified SI: Results Two scenarios: 1.R_0<1. There is only one steady state, i=0, which is stable  disease will die out 2.R_0>1. There are two steady states, i=0, which is unstable, and i=1-1/R_0 which stable  disease will spread R_0 i 1 10

Frequency dependent transmission Infectious encounters are fixed, independent of population size. More appropriate for STIs R_0 is independent of host population size  no endemic threshold wrt N

Underlying interaction of ecology and epidemiology Who is infected? If both, is the disease trophically transmitted? Does infection alter vulnerability to predators? Does infection limit a predators’ ability to catch prey? Does infection alter ability to compete with conspecifics?

Disease assumptions SI disease Density dependent transmission Disease only causes additionally host mortality Disease in predatorDisease in prey

R_0 in PP oscillations All previous work on diseases in oscillatory host use exogenous oscillations, i.e. non-constant parameters. I will use endogenous oscillations (constant parameters) from Rosenzweig—MacArthur model. Will consider 2 models: diseased prey and diseased predator

Rescaling in term of predator— prey—prevalence Disease in predator Disease in prey

Result of rescaling into predator—prey—prevalence Diseased Predator:Diseased Prey: IGP  Food ChainIGP  Exploitative Competition

Invasion criteria at equilibrium For the diseased predator: For diseased prey:

Finding threshold on limit cycle Integrate N and P equations along predator—prey cycle for the period of cycle Consider the infected/prevalence equation over the period of the cycle, assuming that no. of infecteds/prevalence is negligible

Invasion criteria in oscillations For the diseased predator: For diseased prey: For these models and

Disease requires greater transmissibility to become endemic Predator

Prey Disease requires less transmissibility to become endemic

Frequency dependent transmission

Extension: competition Alter prey model such that infected and susceptible prey do not suffer competition equally (c is relative competitiveness of infecteds)

…. with Frequency Dependent transmission For c=1, same as before with FD For c>1, R_0 decreases with host density Disease is endemic as long as For c<1, R_0 increases with host density. Disease is endemic as long as

Summary Endemic criteria depends on time average of host in predator—prey oscillation In our model, oscillations increase endemic threshold in predator ( ) No such pattern for FD Curious case of FD+competition with upper density threshold for endemic disease.

Break

Complex dynamics Using Disease predator model (with DD or FD transmission) Myriad of bistabilities and even a case of tristability Chaos and quasiperiodic dynamics found

Reminder: Standard dynamics FD DD Note: Figures are of prey, disease is in predator.

Bistability via a Cusp bifurcation of limit cycles… Increasing µ=0.5 to µ=0.53 in DD model… Similar pattern occurs in FD model

With increasing µ move from (i) to (vi) 1SS 1SS+1LC 1LC 2LC

Period doubling in FD model » » µ=12 … possibility of 8-cycle

… cascading into chaos Looking at β=µ+0.62, we see a period doubling cascade

Tristability in DD model Saddle-node bif. can occur in DD model  possible endemic SS when < <1 Hopf bif. can move below Saddle-node bif.  there exists a fold—Hopf bif.  possibility of torus bif.

Tristability with Note: & <1 in this region

Tristability with torus

Homoclinic bifurcation? Torus disappears, suspect is collision with saddle limit cycle (a saddle point in Poincaré section)

Homoclinic bifurcation? Torus disappears, suspect is collision with saddle limit cycle (a saddle point in Poincaré section)

Regime shifts Small perturbation results in large change like saddle-node bif. Usually reversible via a long sequence of small perturbations (hysteresis loops) Homoclinic bif. of torus is example of irreversible, once gone, can not return without large perturbation…

Example Reversible(?) Irreversible

Summary Lots of complex dynamics!

Group defending prey Sometimes it is good to be in a crowd… –Large groups can dazzle, confuse or repel predators (be is sight, sound, smell or movement) –Many eyes that improve vigilance –Mob attack enemies

Group defence Similar to diseased prey model, but with explicit competition and growth/death and a Holling IV functional response. Holing IV is Holling II with h=h_0+h_N N

Rewritten for neatness… Where,, (FD) or (DD)

Disease free dynamics We have 4 main scenarios depending on nullclines: 1: Prey only 2: Coexistence 3: Bistability 4: Prey only with transient coexistence

Scenario 4: limit cycle disappears via homoclinic bif.

FD disease Since prevalence equation is independent of prey or predator density, assume it has reached steady state (fix ) and use as bifurcation parameter) System becomes:

What does a disease do? i=0 versus i>0 fixed.

Starting in Scenario 4 and increase i*…

DD disease We can not use same argument as prevalence depends on prey density. This means that Predator—prey—prevalence is a competitive exclusive system… coexistence Instead we use transmissibility as a proxy for prevalence. A similar sequence of Scenarios occurs

Starting in Scenario 4 and increase β…

Coexistence? For DD model, predator—prey—prevalence system is a competitive exclusive system…….. but they do! In fact, in this model, the disease can benefit predators by limiting group defence. Why? Prevalence is self-restricting and can persist at SS for some range (not a point) of prey density. If predators (whose SS require a fixed prey density) can survive in this range, coexistence occurs.

#Work in progress#

Overall Conclusion Predator—prey oscillations can greatly effect disease dynamics. Group defence can be weakened by diseases, possibly helping predators survive #Work in progress# Published parts of talk with Frank Hilker (my supervisor, was in Bath, now in Osnabrück) “Predator—prey oscillations can shift when diseases become endemic” JTB (2013) 316:1-8 “Complex dynamics in an eco-epidemiological model” BMB (2013) 75: “Disease in group-defending prey can benefit predators” Theor. Ecol. (2014) 7: Thank you for listening!