Based on Figures 2 and 3, species one will always shrink in the space above its zero isoclines
and N([alpha] - [theta])/a will always be attracting around abscises axis.
To predict the outcome of competition between insect species within a range of temperatures, we superimpose the zero net growth isoclines
of the species and results show that one species has an equilibrium resource significantly lower than its counterparts.
Since [l.sub.1] and [l.sub.2] are the isoclines
of system (1), [l.sub.1] and [l.sub.2] divide the first octant into several subregions, and the derivative of x and y keeps a fixed sign in each subregion as indicated in Figure 1.
From (20), at point A, on the [dot.k] = 0 isocline
corresponding to n = [bar.n], q = m([k.sub.1]) + a([bar.n]) - ([omega] + [theta])[k.sub.1].
When there is spatial heterogeneity in performance, that is a [greater than] 0, the spatial locality drives the relative position of the isoclines
, and three regions that are qualitatively different may appear.
We have not attempted a formal local-stability analysis of the discrete-grazer model, but the plots of producer productivity and producer isoclines
It is only necessary to determine how the [Mathematical Expression Omitted] and [Mathematical Expression Omitted] isoclines
shift in response to changes in [Alpha] [equivalent to] ([Delta], [Epsilon], [Gamma], k, [Rho], r, [Theta], w), and combine this information with the long-run comparative statics to determine the local comparative dynamics.
For each particular case, this region of initial conditions could be found by means of a system phase portrait (isoclines
) analysis considering the transitions enabling conditions imposed by (20).
In this section, we discuss the intersection of two isoclines
[L.sub.1] and [L.sub.2] by a symmetrical hypothesis and then analyze the dynamics of the model and the global stability of the system.
The interior (positive) equilibria can be evaluated by the intersections of the zero isoclines
The null isoclines
give the threshold of zero growth for each of the two species, and their intersection specifies the equilibrium point.
However, due to the relative position of these isoclines
, [X.sup.*] can be invaded by both lower and higher dispersal mutants, as indicated by vertical arrows.