Download Predators, prey and prevalence

Predators, prey and prevalence
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
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=1-1/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—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 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
• 𝑅0 =
β
𝑚+µ
(predator)
𝑅0 =
β
𝑟+µ
(prey)
• Does not depend on density
• No difference between oscillations and
equilibria
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
1LC
1SS+1LC
2LC
1SS
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!