% cover of reefs - rate of change with constant input & intermittent output I am a novice, not having studied math for years. However, I have a problem that needs a mathematical solution and I hope you can help me. I apologise in advance for using the incorrect language.
Reef ecosystems are in decline worldwide due to bleaching events cause by climate change. The frequency of bleaching events continues to rise, and at some point the death of reefs will be faster than the recovery. I would like to model the % of reef cover over time taking into account the rate of reef recovery (which increases reef cover) and intermittent bleaching events (which decreases reef cover). If the bleaching events weren't intermitted I think I could figure this out but I don't know how to deal with non-continues change.
The factors to be considered are:


*

*RC = Reef cover in percentage (starting at 100%) (y-axis),

*t = time (x-axis),

*B = accumulated bleaching event reef damage (the % of reef damaged by a bleaching event),

*RR = the recovery rate of the reef


So: RCt = RCt0 - Bt + RRt
I am currently calculating RR from empirical data of reef cover X years from a bleaching event. I am just taking the slope of of the change in cover over time.
What I am unsure of is how to calculate B. I would like to be able to change the frequency of bleaching in a given model (every 10 years vs every 5 years) and also the severity of bleaching (% cover lost during each bleaching event). For instance, if we assumed no recovery and bleaching occurred every 5 years with a loss of 10% of RCt0 during each event, 100% of the reef would be lost within 50 years.
I could simply create a continuous slope for B, but the intermittent nature of bleaching events may change the end date of total loss. Also, if I make the model more complex by changing the severity of bleaching and rate of bleaching through time I think keeping it non-continuous may become more important.
The crux of the question is: how do I include the intermittent nature of bleaching events into the model to calculate reef cover over time? 
I hope this question makes sense, if it requires more detail please let me know. I apologise if this isn't the correct place to ask the question, please let me know if there is a more appropriate forum. 
Any help would be greatly appreciated.
Thank you very much,
Adam
 A: The problem you appear to be having is that your terms are not well-enough modeled. Supposing everything is continuous, the differential equation is trivial:
$$\frac{dRC}{dt} = RR(t) - B(t),$$
and has the formal solution
$$RC(t) = RC(t_0) + \int_{t_0}^t(RR(\tau) - B(\tau))d\tau$$
so now you need $RR$ and $B$ as functions of time. You seem to indicate $RR$ is a constant rate so $RR(t) = RR$. Thus
$$RC(t) = RC(0) + RR(t-t_0) - \int_{t_0}^t B(\tau)d\tau.$$
You note however that $B(t)$ is "intermittent" and you need a way to model this. Not knowing anything about your application, I am unable to help you with finding an exact model (maybe someone else is?) but can suggest a few things:
(1) If you model $B$ as being a periodic batch-process--e.g. every $n$ number of years $B$ deletes a certain percentage of the $RC$--you can use this as an effective continuous rate or write an equivalent discrete model where $B$ is an on/off series.
(2) If you model $B$ as being either discretely or continuously stochastic this equation has no nice analytic solution and you should look into doing Monte Carlo simulations. 
