## Cycle of Matter

After a transient dynamics, N and A will tend to one of the two possible equilibria:

(N* = (m + D)h/(v- m- D), A* = I - N*). .. persistence of algae

If v < (m + D)(1 + h/I), the algae will be flushed out of the lake because the reproduction of algae, which is proportional to the consumption of nutrient, is smaller than the loss rate. The resulting equilibrium will be the extinction of algae. The concentration of nutrient in the lake will reach I, the concentration of nutrient in the river. Otherwise, the algae will persist.

The question is: what will be the change in the persistence equilibrium in case the cycle exists versus the case in which decomposition does not occur or is negligible?

We see from  that as the cycling parameter m increases, the equilibrium concentration of nutrient in the lake will increase and the equilibrium of algae will decrease.

However, consider that algae die but are not decomposed (i.e., the term '+ mA' is missing in the first equation of the system ) versus the case when they die but are decomposed. In agreement with intuition, in the second case the concentration of algae will be higher.

Food chain is one of the idealizations of existing ecosystems. Another idealization is a single cycle of matter. Let us consider the simplest cycle with only one trophic level. In order to add to a variety of setups in which ecosystems occur, let us consider a well-mixed lake with a continuous inflow of nutrient, N, and existence of algae, A. The equations may be written as follows:

Parameter D is the flushing of water through the lake (D = inflow = outflow of water divided by the volume of the lake). Parameter I represents a concentration of the nutrient in the incoming river water. Hence, DI is a contribution to the rate of increase of nutrient concentration in the lake. A contribution to the decrease of nutrient from the lake due to outflow is DN. Algae A are being generated through uptake of nutrients and are measured here as the quantity of nutrients in algae. Quantity of nutrients in algae is proportional to the population of algae. Let us assume that algae die with a mortality coefficient m and are flushed out from the lake with a rate DA. Let the nutrient in dead algae be completely decomposed into available nutrients. This will contribute to nutrient concentration in the lake water (mA). 