By Paul Waltman
This ebook makes use of primary principles in dynamical platforms to reply to questions of a organic nature, particularly, questions about the habit of populations given a comparatively few hypotheses concerning the nature in their development and interplay. The central topic handled is that of coexistence below sure parameter levels, whereas asymptotic equipment are used to teach aggressive exclusion in different parameter levels. eventually, a few difficulties in genetics are posed and analyzed as difficulties in nonlinear usual differential equations.
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Extra info for Competition Models in Population Biology
The generality will not be emphasized, but specific situations to be encountered in later sections will be described. The reader who is not interested in the derivations may skip ahead to §2. Section 2 considers a predator populations which feeds differently on different genotypes. Section 3 presents a model of cystic fibrosis, a disease which is believed to be the result of a gene at a single locus. Section 4 presents a parental selection model, an example of a "nonstandard" genetics problem in which "steady states" may have unusual symmetries.
The proof follows from differential inequalities. Because of the lemma we can assume Now, letting 2 = 0, (S = 1 — x — y), one obtains the equation of trajectories in the omega limit set as a two-dimensional system to which our techniques can be applied: There are three critical points which we label £0: (0, 0), £,:(! - A , , 0 ) , E2: (xc, yc), where (xc, yc) satisfies if there is such a solution in the positive quadrant. 1) (the variational matrix) is of the form where At E0 the matrix M takes the form Thus under our hypothesis that A, < 1, E0 is a saddle point whose stable and unstable manifolds lie along the axes.
2). Hence it is an asymptotically stable local attractor and must be the only point of ft. Therefore, we can assume that the ft lies on the boundary of the first quadrant of 2 = 0. If ft contains a point of the y-axis, it must contain the entire trajectory there which is unbounded. Hence any point of the >>-axis is excluded. A similar contradiction follows if ft contains a point of the jc-axis to the right of E\. There remains only to show that E0 and £, do not belong to ft. While this could be done by an appeal to general theorems, we give a direct proof since it is straightforward.