Predators prey and prevalence by Andrew Bate Centre

Predators, prey and prevalence by Andrew Bate Centre for Mathematical Biology, Department of Mathematical Sciences, University of 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 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 N P

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 Births S Natural deaths Infection I Natural deaths Diseaserelated deaths

Simplified SI disease • Assuming population is constant, we can reduce down to one equation and nondimensionalise 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=11/R_0 which stable disease will spread i 1 0 1 R_0

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 predator Disease 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—Mac. Arthur 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 Chain IGP 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

Predator • Disease requires greater transmissibility to become endemic

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 ( < ) and decrease in prey ( > ) • 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

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

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: 2059 -2078 “Disease in group-defending prey can benefit predators” Theor. Ecol. (2014) 7: 87 -100 Thank you for listening!