The role of succession in the evolution of flammability

Abstract

Fire-prone ecosystems contain plants that are both fire-adapted and flammable. It has been hypothesized that these plants were under selection to become more flammable, but it is unclear whether this could be adaptive for an individual plant. We propose arrested succession as a robust mechanism that supports the evolution of flammability in surface fire ecosystems without the need to invoke group selection or additional fitness benefits. We used the natural history of lodgepole pine (Pinus ponderosa) forests, longleaf pine (Pinus palustris) forests, and tall grass prairies to create a general mathematical model of surface fire ecosystems and solved for the evolutionarily stable strategy (ESS) level of flammability. In our model, fires always kill understory plants and only sometimes kill overstory plants. Thus, more flammable plants suffer increased mortality due to fires, but also more frequently arrest succession by clearing their understory of late successional competitors. Increased flammability was selected for when the probability of an overstory plant dying from an individual fire was below a maximum threshold and the rate of succession relative to fires was above a minimum threshold. Future studies can test our model predictions and help resolve whether or not plants have been selected to be more flammable.

Appendix: Methods

Invasion analysis of system without succession

Removing the late successional species yields the following system of equations (Eq. 3):

$$ {\displaystyle \begin{array}{l}\frac{dE}{dt}= cFE-E{\mu \gamma}_RB\\ {}\frac{dP}{dt}= cFP-P{\mu \gamma}_IB\\ {}\frac{dF}{dt}=\mu B\left({\gamma}_RE+{\gamma}_IP\right)- cF\left(E+P\right)\\ {}F=1-E-P\end{array}} $$

To find the ESS level of flammability, we conduct an invasion analysis by testing the stability of the boundary equilibrium in which $\widehat{P}=0$. Unlike in the full system, the reduced system is simple enough that we do not need to assume that $\widehat{F}\approx 0$ and we do not rescale the equations by B. The internal equilibrium values for the resident community are (Eq. 4):

$$ {\displaystyle \begin{array}{l}\widehat{E}=\frac{c-{\mu \gamma}_RB}{c}\\ {}\widehat{F}=\frac{{\mu \gamma}_RB}{c}\end{array}} $$

Evaluated at the internal boundary equilibrium, the Jacobian of Eq. 3 reduces to (Eq. 5):

$$ A=\left[\begin{array}{ccc}\mu B\left({\gamma}_R-{\gamma}_I\right)& 0& 0\\ {}0& 0& c-\mu {\gamma}_RB\\ {}\mu B\left({\gamma}_I-{\gamma}_R\right)& 0& \mu {\gamma}_RB-c\end{array}\right] $$

We note that the boundary condition reduces to a one-dimensional system because E + F = 1. It is straightforward to show that the resident internal equilibrium is stable and that the necessary and sufficient condition for invasion to succeed is (Eq. 6):

$$ {\gamma}_R>{\gamma}_I $$

Thus, in the absence of succession, an invader can only succeed by having a lower γ than the resident. Since we set the minimum value of γ to one, a resident must have a γ_R equal to one to always resist invasion and only an invader with a γ_I equal to one can invade all residents with a different γ. Thus, γ_opt = 1 and a high flammability is always detrimental to an early successional species because it only increases mortality. This is in direct contrast to the result of the model that includes succession, as there is parameter space in which γ_opt > 1. By including succession, high flammability has the added benefit of arresting succession and can potentially offset the losses from increased fire mortality through decreased mortality due to succession.

Invasion analysis of rescaled system

The rescaled system of equations is (Eq. 7):

$$ {\displaystyle \begin{array}{l}\frac{{d E}_1}{d\tau}={c}^{\ast }F\left({E}_1+{E}_2\right)-{E}_1\left({\mu \gamma}_R+{\omega}^{\ast }L\right)+\left(1-\mu \right){\gamma}_R{E}_2\\ {}\frac{{d E}_2}{d\tau}={\omega}^{\ast }{LE}_1-{E}_2\left({\gamma}_R+{s}^{\ast}\right)\\ {}\frac{{d P}_1}{d\tau}={c}^{\ast }F\left({P}_1+{P}_2\right)-{P}_1\left({\mu \gamma}_I+{\omega}^{\ast }L\right)+\left(1-\mu \right){\gamma}_I{P}_2\\ {}\frac{{d P}_2}{d\tau}={\omega}^{\ast }{LP}_1-{P}_2\left({\gamma}_I+{s}^{\ast}\right)\\ {}\frac{d L}{d\tau}={s}^{\ast}\left({E}_2+{P}_2\right)-\mu L\\ {}\frac{d F}{d\tau}=\mu \left({\gamma}_R\left({E}_1+{E}_2\right)+{\gamma}_I\left({P}_1+{P}_2\right)+L\right)-{c}^{\ast }F\left({E}_1+{E}_2+{P}_1+{P}_2\right)\\ {}F=1-{E}_1-{E}_2-{P}_1-{P}_2-L\end{array}} $$

To determine when the invaders, P₁ and P₂, can invade the residents, E₁ and E₂, we test the stability of the boundary equilibrium in which $\widehat{P_1}=0$ and $\widehat{P_2}=0$. We assume that $\widehat{F}\approx 0$ and consequently that $\widehat{E_1}+\widehat{E_2}+\widehat{L}\approx 1$ to solve for the equilibrium values of the residents at the boundary equilibrium. This assumption holds for high values of c^* and is ecologically reasonable in areas where there is very little space open to colonization. The equilibrium values of the resident community (Eq. 8) are:

$$ {\displaystyle \begin{array}{l}\widehat{E_1}=\frac{\mu \left({\gamma}_R+{s}^{\ast}\right)}{\omega^{\ast }{s}^{\ast }}\\ {}\widehat{E_2}=\frac{\mu \left({\omega}^{\ast }{s}^{\ast }-\mu \left({\gamma}_R+{s}^{\ast}\right)\right)}{\omega^{\ast }{s}^{\ast}\left(\mu +s\right)}\\ {}\widehat{L}=\frac{\left({\omega}^{\ast }{s}^{\ast }-\mu \left({\gamma}_R+{s}^{\ast}\right)\right)}{\omega^{\ast}\left(\mu +s\right)}\\ {}\widehat{c^{\ast }F}=\frac{\omega^{\ast}\left({\gamma}_R\mu +{s}^{\ast}\right)+\mu \left({\gamma}_R-1\right)\left({\gamma}_R+{s}^{\ast}\right)}{\gamma_R+{s}^{\ast }+{\omega}^{\ast }}\end{array}} $$

The Jacobian for Eq. 8 can be arranged in upper triangular block form. The eigenvalues, λ, of this matrix are the eigenvalues of the resident community matrix in the absence of invaders and the eigenvalues of the invader community matrix when the invader is rare. It is straightforward to show that in the absence of invaders, the internal equilibrium of the resident community is stable. Thus, the eigenvalues of the resident community matrix are always less than zero. Evaluated at the resident internal equilibrium, the invader matrix (Eq. 9) is:

$$ A=\left[\begin{array}{cc}{c}^{\ast}\widehat{F}-{\gamma}_I\mu -{\omega}^{\ast}\widehat{L}& {c}^{\ast}\widehat{F}+{\gamma}_I\left(1-\mu \right)\\ {}\omega \widehat{L}& -\left({\gamma}_I+{s}^{\ast}\right)\end{array}\right] $$

It is straightforward to show the eigenvalues of the invader matrix are always real. Consequently, for invasion to succeed the dominant eigenvalue of the invader community matrix must be greater than zero. We solve for the boundary surface that separates parameter space into regions of invasion and non-invasion by setting the dominant eigenvalue to zero. This is equivalent to setting the determinant of the invader matrix to zero because the determinant being less than zero is a necessary and sufficient condition for invasion. We solve det(A) = 0, which yields a quadratic function of γ_I with roots (Eq. 10):

$$ {\displaystyle \begin{array}{l}{\gamma}_I={\gamma}_R\\ {}{\gamma}_I={\gamma}_b=\frac{\frac{s^{\ast}\left(\mu \left(1-{s}^{\ast}\right)+{s}^{\ast}\right)}{\mu }{\gamma}_R\mu -{s}^{\ast }{\omega}^{\ast }-\frac{\left({\gamma}_R+{s}^{\ast}\right){\left(\mu +{s}^{\ast}\right)}^2}{\mu \left({\gamma}_R+{s}^{\ast }+{\omega}^{\ast}\right)}}{\mu +{s}^{\ast }}\end{array}} $$

These roots are the values of γ_I that make det(A) = 0 and therefore the dominant eigenvalue exactly zero. When γ_I = γ_R, the invaders and residents are identical, while γ_b represents the value of γ_I that just balances the costs and benefits of γ_I relative to γ_R. Thus, invasion can only occur for the values of γ_I between γ_R and γ_b (Fig. 3). When γ_R = γ_b, there is no value of γ_I that allows invasion and the maximum eigenvalue of zero occurs when γ_I = γ_R. Solving for γ_R when γ_R = γ_b thus gives the ESS value of γ, called γ_opt (Eq. 11; Figs. 2 and 3):

Fig. 3

A necessary and sufficient condition for the dominant eigenvalue to be greater than zero is det(A) < 0. The det(A) = 0 exactly when γ_I equals γ_R or γ_b. In the region between γ_R and γ_b, the det(A) < 0 and invasion is possible. When γ_R = γ_b, there is no value of γ_I that allows for invasion and the best possible invader is the resident. Solving for this value yields the ESS level of flammability, γ_opt

$$ {\gamma}_{\mathrm{opt}}=\frac{\left(\mu +s\right)\sqrt{\mu \left(\mu +4{s}^{\ast }{\omega}^{\ast}\right)}-\mu \left(\mu +{s}^{\ast}\left(2\left({s}^{\ast }+{\omega}^{\ast}\right)+1\right)\right)}{2\mu {s}^{\ast }} $$

For all cases examined, γ_opt is a fitness maximum. Solving for the conditions under which γ_opt > 1 gives the conditions for invasion of pyrophytes (Eq. 12):

$$ {\displaystyle \begin{array}{l}\mu <{\mu}_{\mathrm{crit}},\mathrm{where}\ {\mu}_{\mathrm{crit}}=\frac{s^{\ast }}{4+4{s}^{\ast }}\\ {}{\omega}^{\ast }>\frac{-2\mu {s}^{\ast }-\mu +{s}^{\ast }}{2\mu }-\frac{1}{2}\sqrt{\frac{-4{\mu}^3\left(1+{s}^{\ast}\right)-8{\mu}^2{s}^{\ast 2}-7{\mu}^2{s}^{\ast }-4\mu {s}^{\ast 3}-2\mu {s}^{\ast 2}+{s}^{\ast 3}}{\mu^2{s}^{\ast }}}\\ {}{\omega}^{\ast }<\frac{-2\mu {s}^{\ast }-\mu +{s}^{\ast }}{2\mu }+\frac{1}{2}\sqrt{\frac{-4{\mu}^3\left(1+{s}^{\ast}\right)-8{\mu}^2{s}^{\ast 2}-7{\mu}^2{s}^{\ast }-4\mu {s}^{\ast 3}-2\mu {s}^{\ast 2}+{s}^{\ast 3}}{\mu^2{s}^{\ast }}}\end{array}} $$

The maximum value of μ that allows pyrophytes invasion, called μ_crit, approaches 0.25 as s^* becomes large, creating an upper bound on invasion. The range of ω^* that allows invasion decreases with μ and increases with s^*. The relationship between ω^* and s^* becomes more clear when we analyze γ_opt in new axes corresponding to x = ω^* + s^* and y = ω^* – s^*. We solve $\frac{\partial {\gamma}_{opt}}{\partial y}=0$ for the function y(x) that tracks the value of y that maximizes γ_opt. As x becomes large, the value of y that maximizes γ_opt approaches a small finite value (Eq. 13; Fig. 4):

$$ \underset{x\to \infty }{\lim }y(x)=2\mu $$

Fig. 4

The value of y (the difference between ω^* and s^*) that maximizes γ_opt approaches the value of 2μ as x (the sum of ω^* and s^*) increases. In this figure, μ = 0.1

As x → ∞, the rate of succession increases and γ_opt is maximized when there is a small finite difference between ω^* and s^*. Thus, γ_opt is maximized when ω^* and s^* have a roughly 1:1 ratio, with departures from this ratio decreasing γ_opt and eventually prohibiting invasion by pyrophytes.