Modelling G\(\times\)E effects on consumer-resource interactions
Matt Barbour
2021-07-13
Last updated: 2021-07-13
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Knit directory: foodweb-theory/
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Consumer-resource model
I analyze a continuous time model of a consumer-resource interaction. The resource exhibits logistic growth and the consumer has a type 1 functional response:
\[\frac{dR}{dt}=R(r-\frac{rR}{K}-aC)\] \[\frac{dC}{dt}=C(eaR-m)\]
The following ecological rates govern the dynamics of this model:
r = intrinsic growth rate of resource at low densities
K = carrying capacity of resource
a = attack rate of consumer on the resource
e = conversion efficiency of resources into consumers
m = mortality rate of consumer
I consider these ecological rates as the phenotypes of the consumer and resource. For my initial exploration, I’m going to keep things simple and focus on G\(\times\)E effects of consumers, but I also model the effects of E on resources. I’m also going to link phenotypic change to selection, fitness, and stability.
Following the framework of Gilbert et al. (2014), I assume that the temperature dependence of consumer and resource phenotypes can be modelled as Boltzmann-Arrhenius equations. I focus here on temperature because of its well known influence on consumer and resource phenotypes, and its predictable response to climate change. I also use the biological plausible parameter values that Gilbert et al. (2014) provide in Table 1 and Figure 3 of their paper. I explore the effects of temperature across a gradient from 5-30\(^\circ\)C.
Scaling G\(\times\)E to consumer phenotype
To get a sense for how genotype-by-environment interactions (G\(\times\)E) in consumers might alter these dynamics, I created two different genotypes (A and B). These genotypes have the same “initial” phenotype, which I setup at 15\(^\circ\)C. This choice of temperature was arbitrary and I chose it to follow Fig. 3 of Gilbert et al. (2014). I would argue this represents the effect of genotype (G) as it affects the “intercept” of the phenotype. To simulate a G\(\times\)E effect, I adjusted the activation energy of the temperature relationship (i.e., the slope) assuming a linear tradeoff between genotypes. With these two genotypes, I can visualize G effects (comparing at 15\(^\circ\)C), E effects (mean phenotype change with temperature), and G\(\times\)E effects (different slopes of each genotype).
Scaling G\(\times\)E to consumer attack rate a:
# parameters from Gilbert et al. 2014 and Osmond et al. 2017
E_vC_mean <- 0.46
E_tradeoff <- 0.2
E_vC_A <- E_vC_mean - E_tradeoff
E_vC_B <- E_vC_mean + E_tradeoff
# Genotype A
a_seq <- a_scaling(a0 = a0(a_base = 0.1, v0_C = 1, v0_R = 1,
E_vC = E_vC_A, E_vR = 0.46,
T_C = C_to_K(15), T_R = C_to_K(15)),
v0_C = 1, v0_R = 1, E_vC = E_vC_A, E_vR = 0.46,
T_C = C_to_K(Temp_seq), T_R = C_to_K(Temp_seq))
# Genotype B
a_seq2 <- a_scaling(a0 = a0(a_base = 0.1, v0_C = 1, v0_R = 1,
E_vC = E_vC_B, E_vR = 0.46,
T_C = C_to_K(15), T_R = C_to_K(15)),
v0_C = 1, v0_R = 1, E_vC = E_vC_B, E_vR = 0.46,
T_C = C_to_K(Temp_seq), T_R = C_to_K(Temp_seq)) Scaling G\(\times\)E to consumer mortality rate m:
E_m_mean <- 0.45
# using same tradeoff
E_m_A <- E_m_mean - E_tradeoff
E_m_B <- E_m_mean + E_tradeoff
m_seq <- m_scaling(m0 = m0(m_base = 0.6, E_m = E_m_A, T = C_to_K(15)),
E_m = E_m_A,
T = C_to_K(Temp_seq))
m_seq2 <- m_scaling(m0 = m0(m_base = 0.6, E_m = E_m_B, T = C_to_K(15)),
E_m = E_m_B,
T = C_to_K(Temp_seq))According to Peters (1983), conversion efficiency e is independent of temperature, so I set e=0.15 as in Fig. 3 of Gilbert et al. (2014).
Scaling E to resource phenotype
For resources, I only modelled well known effects of warming.
Scaling intrinsic growth rate r:
# Genotype A
r_seq <- r_scaling(r0 = r0(r_base = 2, E_B = 0.32, T = C_to_K(15)),
E_B = 0.32,
T = C_to_K(Temp_seq))Scaling carrying capacity K:
# Genotype A
K_seq <- K_scaling(K0 = K0(K_base = 100, E_B = 0.32, E_S = 0.9, T = C_to_K(15)),
E_B = 0.32,
E_S = 0.9,
T = C_to_K(Temp_seq))Visualize consumer and resource phenotypes
How will G\(\times\)E effects of temperature alter natural selection?
To answer this question, I have to understand how a small change in the phenotype alters the mean fitness (\(\bar{W}\)) of the consumer or resource population (i.e. directional selection). Importantly, both consumer and resource fitness are density-dependent:
\[\bar{W_R}=\frac{1}{R}\frac{dR}{dt}=r-\frac{r}{K}R-aC\] \[\bar{W_C}=\frac{1}{C}\frac{dR}{dt}=eaR-m\]
To make things easier, I assume the consumer and resource dynamics are at an equilibrium and use these values to to estimate selection on each phenotype. Note that I only assess selection acting on the consumer, since I’m primarily interested in the consequences of G\(\times\)E.
Now I can plot both the G\(\times\)E effects of temperature, but also how these G\(\times\)E alters natural selection:
| Version | Author | Date |
|---|---|---|
| 82fbf95 | mabarbour | 2020-10-15 |
Scaling G\(\times\)E effects to evolutionary change
I’m going to examine how temperature alters the invasion fitness of genotypes A and B. To do this, I need to be able to assess whether Genotype B, e.g., has higher fitness when Genotype A is at equilibrium, and vice versa. But since there phenotypes are different, I need to compare their total phenotype at a particular temperature.
| Version | Author | Date |
|---|---|---|
| 82fbf95 | mabarbour | 2020-10-15 |
Scaling G\(\times\)E effects to food-web stability
I can also look at how the G\(\times\)E effects of temperature alter the stability of the consumer-resource interactions:
Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> =
"none")` instead.
| Version | Author | Date |
|---|---|---|
| 82fbf95 | mabarbour | 2020-10-15 |
While we still observe a typical stability pattern with temperature (explained in detail in Gilbert et al. 2014), we see that G\(\times\)E effects can switch which genotypes confer greater stability depending on temperature.
Reproduce Fig. 3 in manuscript
Reproduce Fig. 3 in manuscript.
| Version | Author | Date |
|---|---|---|
| 82fbf95 | mabarbour | 2020-10-15 |
References
sessionInfo()R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 16.04.7 LTS
Matrix products: default
BLAS: /usr/lib/libblas/libblas.so.3.6.0
LAPACK: /usr/lib/lapack/liblapack.so.3.6.0
locale:
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[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
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[13] workflowr_1.6.2
loaded via a namespace (and not attached):
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