Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications

Figure 7.  First critical values $ \alpha_i^0 $ for bifurcations from the trivial branch (left). Nontrivial branch $ \phi_0^0 $ of the primary transcritical bifurcation at $ \alpha_0^0\approx 1.74 $ for the Beverton-Holt growth-dispersal IDE with right-hand side (5.12), Laplace kernel (3.10) and $ a = 1 $, $ L = 2 $ (center). Total population over $ \alpha\in[0,6] $ along $ \phi_0^0 $ (right)

Figure 10.  First critical values $ \alpha_i^0 $ for bifurcations from the trivial branch (left). Nontrivial branch $ \phi_0^0 $ of the primary transcritical bifurcation at $ \alpha_0^0\approx 1.74 $ for the Beverton-Holt dispersal-growth IDE with right-hand side (5.17), Laplace kernel (3.10) and $ a = 1 $, $ L = 1 $ (center). Total population over $ \alpha\in[0,6] $ along $ \phi_0^0 $ (right)

Figure 15.  First critical values $ \alpha_i^1 $ for bifurcations along the primary branch $ \phi_0^0 $ (left). The $ \alpha $-dependence of the corresponding eigenvalues $ \lambda^1_{i-1}(\alpha) $ for $ \alpha\geq 0.1 $ (center). Period doubling cascade for the Ricker IDE with right-hand side (5.18) and $ \alpha\in[0,40] $ for the Gauß kernel (5.19) with $ a=1 $, $ L=2 $ (right)

Figure 1.  Equilibrium branches $ \phi^\pm(aL) $ as functions of the dispersal parameter $ \alpha=aL $ (left). Four largest eigenvalues $ \lambda^\pm(aL) $ along these two branches of nontrivial solutions to (3.11) (right)

Figure 2.  Subcritical ($ \tfrac{g_{20}}{g_{11}}>0 $) and supercritical ($ \tfrac{g_{20}}{g_{11}}<0 $) fold bifurcation of $ \theta $-periodic solutions to $\left(\Delta_{\alpha}\right)$ described in Thm. 4.2, as well as the exchange of stability between the branches $ \Gamma^+ $ and $ \Gamma^- $ from unstable (dashed line) to exponentially stable (solid) covered in Cor. 4.3

Figure 3.  Transcritical bifurcation of $ \theta $-periodic solutions to $\left(\Delta_{\alpha}\right)$ from a branch $ \Gamma_1 $ into $ \Gamma_2 $ described in Prop. 4.6, as well as the exchange of stability from unstable (dashed line) to exponentially stable (solid)

Figure 4.  Subcritical $ (\bar g/g_{11}>0) $ and supercritical $ (\bar g/g_{11}<0) $ pitchfork bifurcation of $ \theta $-periodic solutions to $\left(\Delta_{\alpha}\right)$ from a branch $ \Gamma_1 $ into $ \Gamma_2 $ described in Prop. 4.7, as well as the exchange of stability from unstable (dashed line) to exponentially stable (solid)

Figure 5.  Branch of the subcritical fold bifurcation for $\left(\Delta_{\alpha}\right)$ with right-hand side (5.2) and kernel (5.1) (left). Total population over $ \alpha\in[0.0,0.3] $ with $ a = \tfrac{1}{4} $, $ L = 2 $ along the branch (right)

Figure 6.  Schematic bifurcation diagrams for the cosine kernel (5.1) with $ aL\leq\tfrac{1}{2} $ illustrating branches of $ \theta $-periodic solutions: Ex. 4 has a subcritical fold bifurcation of fixed points at $ (\phi^\ast,\alpha^\ast) $ (left). After a supercritical period doubling at $ (0,\alpha_0^0) $, Ex. 5 has a supercritical pitchfork bifurcation of $ 2 $-periodic solutions at $ (\phi_\pm(\alpha_1^0),\alpha_1^0) $ (right). Fixed point branches are solid $ (\theta = 1) $, branches of $ 2 $-periodic solutions are dashed $ (\theta = 2) $ and $ 4 $-periodic solutions are indicated by dotted lines $ (\theta = 4) $

Figure 8.  Branch $ \phi_1^0 $ of the supercritical pitchfork bifurcation at $ \alpha_1^0\approx 5.12 $ for the Beverton-Holt growth-dispersal IDE with right-hand side (5.12), Laplace kernel (3.10) and $ a = 1 $, $ L = 2 $ (left). Total population over $ \alpha\in[\alpha_1^0,7] $ along $ \phi_1^0 $ (right)

Figure 9.  Branch $ \phi_2^0 $ of the transcritical bifurcation at $ \alpha_2^0\approx 12.73 $ for the Beverton-Holt growth-dispersal IDE with right-hand side (5.12), Laplace kernel (3.10) and $ a = 1 $, $ L = 2 $ as part of a fold (left). Total population over parameters $ \alpha\in[12.4, 15] $ reflecting a fold in $ \phi_2^0 $ (right)

Figure 11.  Branch $ \phi_1^0 $ of the supercritical pitchfork bifurcation at $ \alpha_1^0\approx 5.12 $ for the Beverton-Holt dispersal-growth IDE with right-hand side (5.17), Laplace kernel (3.10) and $ a = 1 $, $ L = 2 $ (left). Total population over $ \alpha\in[5,6] $ along $ \phi_1^0 $ (right)

Figure 12.  Branch $ \phi_2^0 $ of the transcritical bifurcation at $ \alpha_2^0\approx 12.73 $ for the Beverton-Holt dispersal-growth IDE with right-hand side (5.17), Laplace kernel (3.10) and $ a=1 $, $ L=2 $ as part of a fold (left). Total population over parameters $ \alpha\in[12,13] $ reflecting a fold in $ \phi_2^0 $ (right)

Figure 13.  Schematic bifurcation diagrams: Branches of $ \theta $-periodic solutions in the Beverton-Holt IDE having the right-hand sides (5.12)/(5.17), $ \alpha\in[0,50] $ and the Laplace kernel (5.19) with $ a=1 $, $ L=2 $ (left), as well as IDE with right-hand side (5.18) and $ \alpha\in[0,3000] $ (logarithmic axis) for the Gauß kernel (5.19) with $ a=1 $, $ L=2 $ (right). Fixed point branches are solid $ (\theta=1) $, branches of $ 2 $-periodic solutions are dashed $ (\theta=2) $ and $ 4 $-periodic solutions are indicated by dotted lines $ (\theta=4) $

Figure 14.  First critical values $ \alpha_i^0 $ for bifurcations from the trivial branch (left). The $ \alpha $-dependence of the corresponding eigenvalues $ \lambda_{i-1}^0(\alpha) $ (center). Primary transcritical bifurcation of the branch $ \phi_0^0 $ at $ \alpha^0_0\approx 1.36 $ for the Gauß kernel (5.19) with $ a=1 $, $ L=2 $ (right)