Pest control by generalist parasitoids: A bifurcation theory approach

Figure 1.  Graphs of the host nullcline, $ F(N) $, drawn in a thick solid curve and the parasitoid nullcline, $ G(N) $, drawn in a thin solid curve for model (2.1) with the Holling type Ⅱ functional response

Figure 2.  The number of coexistence equilibria, denoted by $ E^* $, in different regions of the $ K_1K_2 $-plane. $ \mathcal{C}_1 $ (thick solid curve) and $ \mathcal{C}_2 $ (thin solid curve) indicate branches of saddle-node bifurcations and the dot-dashed line indicates a branch of transcritical bifurcations. In the white regions, no coexistence equilibria exist and the prey-extinction equilibrium $ E_{K_2} $ is globally asymptotically stable. See (2.11) for $ \theta_{1, 2} $ and $ \kappa $

Figure 3.  Phase portrait analysis when (a) $ K_{1}<1/b $ and (b) $ K_{1}>1/b $ and there are no coexistence equilibria, demonstrating the global stability of the host-free (or pest-free) equilibrium $ E_{K_{2}} = (0, K_{2}) $

Figure 4.  Two-parameter bifurcation diagram in the $ K_1K_2 $-plane. The parameter values used here are $ a = 0.4, b = 8, r_1 = 2, r_2 = 0.04, $ and $ \gamma = 10 $. A thick (thin) solid curve represents a branch of super-critical (sub-critical) Hopf bifurcations, denoted by $ H^+ $ ($ H^- $). Curves of fold bifurcations (FB) are shown as dotted curves. The dot-dashed line indicates the line of transcritical bifurcations (TB). The cyclic-fold bifurcations (CFB) is shown as a dotted curve. The curve of homoclinic bifurcations is depicted as a thick grey solid curve and the neutral saddle curve is the dot-dashed curve. In each region of the bifurcation diagram, the symbol $ {\bf R}(i, j) $ shows the number of equilibria ($ i = 3, 4, 5, 6 $) and the number of limit cycles ($ j = 0, 1, 2 $). When there is a unique limit cycle, the stability of the limit cycle is indicated by a superscript: S for stable and U for unstable

Figure 5.  Representative phase portraits and a caricature of the bifurcation diagram in Figure 4, emphasizing different dynamics in the different regions separated by the bifurcation curves. In each phase portrait, open circles indicate unstable equilibria and filled circles denote stable equilibria. The labels in the figure legend have the same meaning as in Figure 4

Figure 6.  Representative phase portraits of system (2.1) on the bifurcation curves between the regions in Figure 5, that involve a homoclinic orbit. The open and closed circles have the same interpretation as in Figure 5

Figure 7.  (a)-(d) Phase portraits for model (2.1). Host nullclines (thick solid curves) and parasitoid nullclines (dashed curves). $ K_2 = 1.14408169 $ and in (a) $ K_1 = 0.3 $, (b) $ K_1 = 0.6 $, (c) $ K_1 = 0.871 $, and (d) $ K_1 = 0.9 $. All of the other parameters are the same as in the two-parameter bifurcation diagram shown in Figure 4. Equilibria can be found at the intersection of the host and parasitoid nullclines. Filled circles indicate stable equilibria and open circles indicate unstable equilibria. SLC and ULC indicate stable and unstable limit cycles, respectively. (e) One-parameter bifurcation diagram with $ K_1 $ as the bifurcation parameter. Branches of stable equilibria are drawn using thick solid curves and unstable equilibria are drawn using thin solid curves. The largest and smallest values of the $ N $-coordinate of the orbitally asymptotically stable limit cycles are shown using filled circles and of the unstable limit cycles using open circles. $ H^+ $ and $ H^- $ indicate super- and sub-critical Hopf bifurcations, respectively. CFB indicates the cyclic-fold bifurcation (or saddle-node bifurcation of limit cycles)

Figure 8.  (a)–(d) Phase portraits for model (2.1): $ K_2 = 1.3 $. All of the other parameters are the same as in Figures 4 and 7, except (a) $ K_1 = 0.8 $, (b) $ K_1\approx 1.1474041 $, (c) $ K_1 = 1.5 $, and (d) $ K_1 = 2.5 $. Nullclines curves and equilibria are indicated as in Figure 7. (e) One-parameter bifurcation diagram: HC and FB indicate homoclinic and fold bifurcations, respectively. Bifurcation curves are indicated and labelled as in Figure 7

Figure 9.  An example illustrating the missing case in [13]. Consider the following parameters: $ a = 0.5, b = 0.5, r_1 = 8, r_2 = 2, \gamma = 1.1, K_1 = 5, $ and $ K_2 = 16.2 $. The graph of the host, $ F(N) $, (leaf miners) nullcline is drawn as a solid thick curve and the parasitoid, $ G(N) $, nullcline is a solid thin curve. The maximum of $ F(N) $ occurs at $ \frac{1}{2}(K_2-\frac{1}{b}) = 1.5 $. Equilibria can be found at the intersections of the host and parasitoid nullclines. There are two coexistence equilibria even though $ K_2>r_1/a = 16 $ and $ B = G(1.5)\approx 20.01857>19.6 = F(1.5) = A $. $ F(N) $ and $ G(N) $ intersect at 0.4513580760 and 1.067501870. Both of these points lie to the left of the maximum of $ F(N) $. The corresponding coexistence equilibria $ (N_i^*, P_i^*), \ i = 1, 2 $ are $ (1.0675, 19.3) $ and $ (0.4514, 17.84) $, respectively

Figure 10.  Discrepancy (hatched region) between the graphs of $ A = B $ in [13] and the portion of the fold curve, $ \mathcal{C}_1 $, above the curve of transcritical bifurcations (TB) in the two-parameter bifurcation diagram. The parameter values are the same as the ones used in Figure 9. Magal et al. [13] claim that there are no interior equilibria in the hatched region. However, there are actually two interior equilibria in that region

Figure 11.  The number of $ E^* $ in different regions on the $ K_1K_2 $-plane, using Descartes' rule of signs in cases, (a) $ \gamma\le r_2 b/(2a) $, (b) $ r_2b/(2a)<\gamma <r_2 b/a $, and (c)$ \gamma \ge r_2b/a $. $ Z_1 (K_1) $ is defined in (A.3b) and $ \kappa $ in (A.4). There are no $ E^* $ in the white regions

Figure 12.  Possible graphs of $ L(N) = N^3+AN^2+BN $ for $ N>0 $