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Appendix A. A description of the cold shock model.

This Appendix describes the modification of the model to incorporate a cold shock. In the text we simulated a cold shock in which a Batrachochytrium dendrobatidis population at 23ºC experiences a 1-week long decrease in temperature to 10ºC before being brought back to 23ºC. It would, however, be possible to use the model to simulate the population growth in any fluctuating temperature environment (as long as the temperature-dependent parameters are defined).

In order to incorporate a fluctuating temperature environment, we discretize the model into hourly time steps. The temperature for each hour is set to be either 23ºC for most of the simulation, or 10ºC during the cold shock. The model still divides the population of B. dendrobatidis into a mobile infectious zoospore stage, Z, and a sessile developing sporangium stage, S. However, the sporangium stage is divided into a number (S_max) of 1 hour long sub-stages or cohorts, S₁ to S_Smax. For each hourly time step, the fraction exp {-d_Z} of the zoospores survive, and the fraction exp {-m_Z} of the surviving zoospores encyst and enter the youngest sporangium cohort, S₁. The remaining (1-exp{- m_Z } ) of the surviving zoospores remain in the zoospores class. For each time step, d_Z and m_Z are set to the values for either 10ºC or 23ºC, depending on the current temperature.

Each hour the individuals in each cohort, S_i, mature into the S_i+1 cohort. For each cohort the model keeps track of a maturation index and a fecundity index for each cohort. The maturation index starts at zero at the point of encystment, and for each hourly time-step, 1/ is added to the index. The fecundity index also starts at zero for the first cohort, and for each hourly time step F/ is added to the index. In each case,and F are set to the values for either 10ºC or 23ºC, depending on the current temperature. When the maturation index for a cohort reaches or exceeds 1 during a time step, the sporangia in that cohort become zoospore-producing sporangia. At the end of that time step, the zoospore-producing sporangia in that cohort release their zoospores according to the fecundity index for that cohort, and that cohort is removed from the sporangium stage.

Simulations of the discretized version of the model with the parameters set to constant values (either those for 10ºC or 23ºC) are in good agreement with the corresponding simulation from the delay-differential equation (maximum discrepancy between simulation trajectories 2% over the first 50 d of the simulation). For the cold shock trajectory in Fig. 6 of the text, the temperature is 23ºC for the first 400 hours, reduced to 10ºC for the next week, and then brought back up to 23ºC for the remainder of the simulation. For the sporangia present during the cold shock, the algorithm described above results in maturation times and fecundities for each cohort that are the weighted averages of the parameter values at 10ºC and 23ºC, with the weighting depending on the number of hours spent at each of the two temperatures.