Fire mitigates bark beetle outbreaks in serotinous forests

Bark beetle outbreaks and forest fires have imposed severe ecological damage and caused billions of dollars in lost resources in recent decades. The impact of such combined disturbances is projected to become more severe, especially as climate change takes its toll on forest ecosystems in the coming years. Here, we investigate the impact of multiple disturbances in a demographically heterogeneous tree population, using an age-structured difference equation model of bark beetle outbreaks and forest fires. We identify two dynamical regimes for beetle and fire dynamics. The model predicts that fire helps dampen beetle outbreaks not only by removing host trees but also by altering the demographic structure of forest stands. We show that a stand thinning protocol, which reduces the population size of the largest few juvenile classes by a small percentage, is able to significantly reduce beetle-induced tree mortality. Our research demonstrates one approach to capturing compound disturbances in a mathematical model. Electronic supplementary material The online version of this article (10.1007/s12080-021-00520-y) contains supplementary material, which is available to authorized users.

1 Derivation of Model Equations j n+1,1 = dJ n + I n−2 + F n (1) New juvenile trees are created each year according to equation 1. The number of juveniles of age 1 is equal to the total number of juveniles that died last year, dJ n = d N k=1 j n,k , plus the number of grey snags I −2 , plus the number of burnt snags (trees that burned the previous summer) F n .
As mentioned in the main text, we use P n to be the severity of forest fire in year n. It is defined by an exponentially decaying sum over the unburnt forest area from previous years, so P n is not a state variable.
The growth of juvenile trees is defined by equation 3. A fraction 1 − d of juveniles from class k − 1 grows into class k juveniles, minus the trees in this class that burn, proportional to P n .
The number of susceptibles in the spring of year n + 1 is equal to the number of susceptibles in the spring of year n, plus the number of juveniles growing into mature trees ((1 − d)jn, K). We subtract the trees that were infested in the summer of year n, I n , the infested and burnt trees, α3 T P n I n , the trees that were only burnt, α2 T P n S n , and the juveniles from class K that would have become mature if they had not caught fire, ( α2 T P n (1 − d)j n,K ).
S n+1 = S n + (1 − d)j n,K − (I n + α 3 T P n I n ) − α 2 T P n (S n + (1 − d)j n,K ) (4) 5) The model for infested trees is based on ricker-style dynamics, where r 1 is the reproduction rate of beetles, and the exponential term denotes the probability that each individual will find a susceptible tree. The number of infested trees burned is subtracted after reproduction, for simplicity. As in Duncan et al., we have I n+1 = r 1 I n e In+In−1+Fn+Jn+1 , where the exponent is the number of current non-susceptible trees. From the conservation of tree-equivalents The burnt snags in the spring of year n+1 is equal to the sum of the number of burnt juveniles ( α1 T P n K−1 i j n,K ), susceptibles ( α2 T P n (Sn + (1 − d)j n,K )), and infested ( α3 T I n P n ).
The following lemma demonstrates that the model equations preserve the total tree equivalent population present in the initial conditions. Lemma 1 Let I −1 + I −2 + F 0 + J 0 + S 0 = T . The equation I n−1 + I n−2 + F n + J n + S n = T is true for all n ≥ 0 under the evolution equations 7a-7f.
Proof First, notice that the only individuals leaving the Juvenile compartment J n are the surviving oldest juvenile age class (1 − d)j n,K and the sum of the trees burned from each juvenile age class (except the oldest), α1 T P n K−1 k=1 j n,k . The individuals entering the Juvenile compartment are the seedlings germinating in the canopy gaps created by the gray snags I n−2 and the trees burned the previous summer, F n . Therefore we have the following equation for the total number of juvenile trees J n+1 .
For the inductive step, assume that I n−1 + I n−2 + F n + J n + S n = T is true, then we have: where we use the definition of F n+1 .
where we obtain the conservation equation for the spring of year n + 1 by definition of S n+1 . Therefore the inductive step is true, and the model equations conserve tree equivalents.

Initial conditions
The initial conditions we use in the model are defined as follows.
These conditions were chosen to provide reasonable representations of the system being modeled. The model appears to be robust to the choice of initial conditions and so the observations in the main text hold for any reasonable set of initial conditions. Note that I −1 + I −2 + F 0 + J 0 + S 0 = 110000 = T , where T is the parameter that determines the total number of stems, from the main text.