Trade-off between tree defenses and reproduction
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Trade-off between tree defenses and reproduction
Valeria Aschero
Natacha Chacoff
Silvina Velez
Valdemar Delhey
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Biological background
Herbivores
Induced responses (spines)
Reproduction (fruits)
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Design:•Prosopis flexuosa•Reserve and cattle grazed sites•Response variable=Fruit production/ind•Explanatory variable= Spine length
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Looking at our data
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Imagine the deterministic model
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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)
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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
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Negative Exponential Fit
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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")
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Plot hyperbolic model
Which one do you vote?...doodle.com/espina$#@!%$#@!
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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 )
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Confidence intervals of both models (NegExp vs Hyperbolic)
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Comparing models
The Hyperbolic is marginally better:
Δ AIC= 0.8
But…
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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")
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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!
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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
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