Trade-off between tree defenses and reproduction

Valeria Aschero

Natacha Chacoff

Silvina Velez

Valdemar Delhey

Biological background

Herbivores

Induced responses (spines)

Reproduction (fruits)

Design:•Prosopis flexuosa•Reserve and cattle grazed sites•Response variable=Fruit production/ind•Explanatory variable= Spine length

Looking at our data

Imagine the deterministic model

Eyeball estimates of the Negative Exponential parameters (estimando a ojo)

Deterministic model:

#fruits = a e -b spine_length

Stochastic model:Negative Binomial(counts, variance higher than mean)

NUMERICAL OPTIMIZATION USING mle2()

fEnBn=function(a,b,k){media=a*exp(-b*espi$spine)-sum(dnbinom(espi$fruits,mu=media,size=k,log=T)) }

m1 = mle2(fEnBn, list(a=750,b=0.1,k=1), data=espi, method="Nelder-Mead")

GENERALIZED LINEAR MODEL (negative binomial)

Linearized function (log link):Log(# fruits) = Log(a) –b spine_length

glmnb=glm.nb(fruits~spine, data=espi)

Both approaches yielded equal estimates

Fitting the Negative Exponential Model

Negative Exponential Fit

Fitting the Hyperbolic model with mle2()

Hyperbolic model:

Fruits= a / (b + spine length)

fHyBn=function(a,b,k){ media=a/(b+espi$spine) -sum(dnbinom(espi$fruits,mu=media,size=k,log=T))

}

mHy= mle2(fHyBn, list(a=3000,b=10,k=1), data=espi, method="Nelder-Mead")

Plot hyperbolic model

Which one do you vote?...doodle.com/espina$#@!%$#@!

Estimating CI for the hyperbolic model

# Generar valores aleatorios de parámetros usando matriz de varianza y covarianza

coefazHy=rmvnorm(1000,coef(mHy),vcov(mHy))sec.esp=seq(0.05,40,leng=100)

curvasHy=NULLfor(i in 1:length(sec.esp)){ temp2=coefazHy[i,1]/(coefazHy[i,2]+sec.esp) curvasHy=cbind(curvasHy,temp2)}

cinfHy=apply(curvasHy,1,quantile, prob=0.025 )csupHy=apply(curvasHy,1,quantile, prob=0.975 )

Confidence intervals of both models (NegExp vs Hyperbolic)

Comparing models

The Hyperbolic is marginally better:

Δ AIC= 0.8

But…

Using the Neg. Exp. model: Are parameters different in the reserve and cattle grazed sites?

“a” PARAMETERS DIFFERENT :glmnb.a<-glm.nb(fruits~spine+situation,data=espi)

fEnBntA=function(aC,aR,b,k){ a=c(aC,aR)[espi$situation]; media=a*exp(-b*espi$spine) -sum(dnbinom(espi$fruits,mu=media,size=k,log=T)) }mtA = mle2(fEnBntA, list(aC=375,aR=375,b=0.1,k=1), data=espi, method="Nelder-Mead")

“b” PARAMETERS DIFFERENT:glmnb.b<glm.nb(fruits~spine+spine:situation,data=espi)

fEnBntB=function(a,bC,bR,k){ b=c(bC,bR)[espi$situation]; media=a*exp(-b*espi$spine) -sum(dnbinom(espi$fruits,mu=media,size=k,log=T)) }mtB = mle2(fEnBntB, list(a=362,bC=-0.09,bR=0.09,k=1), data=espi, method="Nelder-Mead")

Is it worthy to use different parameters for trees in the reserve and in cattle grazed sites?

LRTest Results:

“a” equals vs “a” different

df=1, LRstat= 0.18, p=0.66

“b” equals vs “b” different

df=1, LRstat= 0.04 , p=0.83

NO!

To take home:

Fruit production per individual decreases with spine

length

Negative exponential and hyperbolic model both could

be used to describe the response

We don't have enough evidence to say that the

relationship between fruits and spine length differ

between protected and cattle grazed sites