Biomanipulation Biomanipulation Biomanipulation is the deliberate alteration of an ecosystem by adding or removing species, especially predators. Changing the fish population of bodies of water as a part of watershed management can facilitate desirable changes in aquatic ecosystems suffering from eutrophication characterized by domination by phytoplankton aiding ecosystem restoration, an application of restoration ecology. In ponds or lakes alternative stable conditions, one with high algae populations, little other plant life, and turbid water, another with low algae populations, a diverse plant population, and clear water, may exist. In addition to prevention of excess nutrients such as phosphorus and nitrates, removal of
Proofs involving the addition of natural numbers induction on "a", and so we have proved the base case "b" = 1. Now, suppose that for all natural numbers "a", we have "a" + "b" = "b" + "a". We must show that for all natural numbers "a", we have "a" + "S"("b") = "S"("b") + "a". We have This completes the induction on "b". Proofs involving the addition of natural numbers This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity. These proofs are used in the article Addition of natural numbers. This article will use the