Uni-directional Interaction and Plant–Pollinator–Robber Coexistence

Abstract

A mathematical model for the plant–pollinator–robber interaction is studied to understand the factors leading to the widespread occurrence and stability of such interactions. In the interaction, a flowering plant provides resource for its pollinator and the pollinator has both positive and negative effects on the plant. A nectar robber acts as a plant predator, consuming a common resource with the pollinator, but with a different functional response. Using dynamical systems theory, mechanisms of species coexistence are investigated to show how a robber could invade the plant–pollinator system and persist stably with the pollinator. In addition, circumstances are demonstrated in which the pollinator’s positive and negative effects on the plant could determine the robber’s invasibility and the three-species coexistence.

Appendices

Appendix A

Proof of Theorem 3.2

• (i) Since α ₂₁<d ₂(d ₁+ar ₁)/r ₁, we obtain a ₂>0 and P ₁ is locally asymptotically stable. Thus, if −a ₁≤0 or Δ ₁<0, it follows from (5) that there is no positive root. By Proposition 3.1, P ₁ is globally asymptotically stable.

• (ii) It follows from a ₁<0 and Δ ₁>0 that there are two positive equilibria $P_{12}^{-}$ and P ₁₂. By (3), we obtain tr(J(P ₁₂))<0. A long but straightforward computation shows that

  $$\det J(P_{12}) = \frac{d_2 N_2^+}{ b(\alpha_{21} -ad_2)^2 } \varLambda,\quad \varLambda= \alpha_{21} a_0 \bigl(1+bN_2^+ \bigr)^2 - \alpha_{12} b. $$

  By (5), we have

  $$\varLambda = b \alpha_{21} (2a_0-ba_1) \biggl(N_2^+ - \frac{2ba_2-B}{2a_0-ba_1}\biggr). $$

  It follows from a ₁<0 that

  $$2a_0-ba_1 = b\biggl(\frac{\alpha_{12} }{\alpha_{21}} + \frac{\beta_1+br_1}{ \alpha_{21} -ad_2}\biggr) >0,\qquad 2ba_2-a_1 = \frac{\alpha_{12} }{\alpha_{21}} -\frac{\beta_1+br_1}{ \alpha_{21} -ad_2}>0. $$

  Since

  $$(2a_0-ba_1) (-a_1+\sqrt{ \varDelta _1}) -2a_0(2ba_2-a_1) = (2a_0-ba_1) \sqrt{\varDelta _1} +b \varDelta _1>0, $$

  then

  $$N_2^+ = \frac{-a_1+\sqrt{\varDelta _1}}{2a_0} > \frac{2ba_2-a_1}{2a_0-ba_1}. $$

  Thus, we have detJ(P ₁₂)>0, which implies P ₁₂ is locally asymptotically stable.

  While $b^{2} \varDelta _{1} - (2a_{0}-ba_{1})^{2} = -4a_{0}^{2} +4ba_{0}a_{1} -4b^{2}a_{0}a_{2}<0$, we obtain $b\sqrt{\varDelta _{1}} - (2a_{0}-ba_{1})<0$, and

  

  Hence, we have

  $$N_2^- = \frac{-a_1-\sqrt{\varDelta _1}}{2a_0} < \frac{2ba_2-a_1}{2a_0-ba_1}. $$

  Thus, $\det J(P_{12}^{-})<0$, which implies $P_{12}^{-}$ is a saddle point. By Proposition 3.1, (ii) is proved.

• (iii) It follows from B<0 and Δ ₁=0 that there exists a unique positive equilibrium P ₁₂, which is the coincide of equilibria $P_{12}^{-}$ and P ₁₂. By the criterion for saddle-node points (Theorem 7.1, Zhang et al. 1992), P ₁₂ is a saddle-node point. By Proposition 3.1, (iii) is proved.

• (iv) It follows from α ₂₁>d ₂(d ₁+ar ₁)/r ₁ that C<0 and P ₁ is a saddle. By (5), there is at most one interior equilibrium $(N_{1}^{+}, N_{2}^{+})$. It follows from Proposition 3.1 that there exists a unique interior equilibrium $P_{12}(N_{1}^{+}, N_{2}^{+})$ of system (2), which is globally asymptotically stable in the positive quadrant.

 □

Appendix B

Proof of Theorem 4.2

By r ₁/d ₁>max{d ₂/(α ₂₁−ad ₂),cd ₃/(α ₃₁−d ₃)}, we obtain $\mu_{2}^{(1)}>0$ and $\mu_{3}^{(1)}>0$. Thus, E ₁ is a saddle point, and equilibria E ₁₂ and E ₁₃ exist. Since $\mu_{2}^{(1)}>0$, $E_{12}(N_{1}^{+},N_{2}^{+},0 )$ is globally asymptotically stable in the interior of the (N ₁,N ₂)-plane (Theorem 3.2). We show $\mu_{3}^{(12)} >0$ as follows. By the second equation of (2), we have

$$-d_2 + \frac{\alpha_{21}N_1^+}{1 + a N_1^+}>-d_2 + \frac{\alpha_{21}N_1^+}{1 + a N_1^+ + b N_2^+}=0 $$

then

$$-d_2 + \frac{\alpha_{21}N_1^+}{1 + a N_1^+}> -d_2 + \frac{\alpha_{21}N_1^\#}{1 + a N_1^\#}. $$

By the monotonicity of function N ₁/(1+aN ₁) that $N_{1}^{+} > N_{1}^{\#}$. It follows from the monotonicity of function N ₁/(c+N ₁) that

$$\mu_3^{(12)} = -d_3 + \frac{\alpha_{31}N_1^+}{c + N_1^+} > -d_3 + \frac{\alpha_{31}N_1^\#}{c + N_1^\#} =0 $$

which implies $\mu_{3}^{(12)} >0$, i.e., E ₁₂ is unstable in the N ₃-axis direction.

When $\alpha_{21} < d_{2}(a + 1/N_{1}^{\#})$, we obtain $\mu_{2}^{(13)} <0$. Then there is no positive equilibrium of (1). Indeed, suppose $E^{*}(N_{1}^{*},N_{2}^{*}, N_{3}^{*})$ is a positive equilibrium of (1). By the third equation of (1), we obtain $N_{1}^{*}=N_{1}^{\#}$. It follows from (13) that

$$-d_2 + \frac{\alpha_{21}N_1^*}{1 + a N_1^* + b N_2^*} < -d_2 + \frac{\alpha_{21}N_1^\#}{1 + a N_1^\#} = \mu_2^{(13)}<0 $$

which forms a contradiction by the second equation of (1).

• (i) It follows from $\mu_{3}^{(12)}>0$ that E ₁₂ is a saddle point: the (N ₁,N ₂)-plane is the stable manifold and the N ₃-axis is the unstable manifold. Since $\mu_{2}^{(13)} >0$ and condition (9) holds, E ₁₃ is a saddle point: the (N ₁,N ₃)-plane is the stable manifold and the N ₂-axis is the unstable manifold. Thus, the boundary equilibria can not form a heteroclinic cycle. It follows from the acyclicity theorem of Butler et al. (1986) that uniform persistence of (1) is guaranteed.

• (ii) It follows from $\mu_{2}^{(13)} >0, \mu_{2}^{(\phi)} >0$ and (10) that, E ₁₃ is unstable; E _ϕ is also unstable: the (N ₁,N ₃)-plane is the stable manifold, and the N ₂-axis is the unstable manifold. Thus, the boundary equilibria and periodic orbits cannot form a heteroclinic cycle. Similar to the proof of (i), (ii) is proved.

• (iii) It follows from $\mu_{2}^{(13)} <0$ and (9) that E ₁₃ is locally asymptotically stable in $R_{+}^{3}$. Let ϖ ₁₃ be the basin of attraction of E ₁₃ in ϖ. Then ϖ ₁₃ is open and forward invariant, and ϖ−ϖ ₁₃ is closed and forward invariant in ϖ. Suppose the interior of ϖ−ϖ ₁₃ (i.e., int(ϖ−ϖ ₁₃)) is not empty. Since the boundary equilibria O,E, and E ₁₂ are hyperbolic saddle points and cannot form a heteroclinic cycle, hypotheses of (H-1) to (H-4) derived by Butler et al. (1986) are satisfied in ϖ−ϖ ₁₃. Thus, system (1) restricted on ϖ−ϖ ₁₃ is uniformly persistent and has a positive equilibrium E ^∗ as a result of Butler et al. (1986). This forms a contradiction since there is no positive equilibrium when $\mu_{2}^{(13)} <0$. Thus, int(ϖ−ϖ ₁₃) is empty and (iii) is proved.

• (iv) It follows from $\mu_{2}^{(\phi)} <0$ and (10) that E _ϕ is locally asymptotically stable. Let ϖ _ϕ consist of basins of attraction of E ₁₃ and E _ϕ in ϖ. Similar to the proof in (iii), int(ϖ−ϖ _ϕ ) is empty and (iv) is proved.

• (v) It follows from $\mu_{2}^{(13)} >0$ and $\mu_{2}^{(\phi)} <0$ that E _ϕ is locally asymptotically stable in ϖ. When $\mu_{2}^{(13)} <0$ and $\mu_{2}^{(\phi)} >0$, E ₁₃ is stable in the N ₂-axis direction, and there is a positive solution of (1) which converges to E ₁₃. Thus, system (1) is not persistent.

 □

Appendix C

Proof of Theorem 4.3

It follows from r ₁/d ₁>d ₂/(α ₂₁−ad ₂) and r ₁/d ₁<cd ₃/(α ₃₁−d ₃) that $\mu_{2}^{(1)}>0$ and $\mu_{3}^{(1)}<0$. Then E ₁ is globally asymptotically stable in the interior of the (N ₁,N ₃)-plane, and is unstable in the N ₂-axis direction; E ₁₂ is globally asymptotically stable in the interior of the (N ₁,N ₂)-plane.

• (i) It follows from $\alpha_{31} > d_{3}(1 + c/N_{1}^{+})$ that $\mu_{3}^{(12)} >0$. Then the equilibrium $E_{12}(N_{1}^{+}, N_{2}^{+},0)$ is unstable in the N ₃-axis direction. Since the boundary equilibria O,E ₁, and E ₁₂ are hyperbolic and cannot form a heteroclinic cycle, system (1) is uniformly persistent as a result of Butler et al. (1986).

• (ii) It follows from $\alpha_{31} < d_{3}(1 + c/N_{1}^{+})$ that $\mu_{3}^{(12)}<0$ and $-d_{3} +\alpha_{31} N_{1}^{+} /(c+ N_{1}^{+})<0$. Then E ₁₂ is locally asymptotically stable in $R_{+}^{3}$. Let ϖ ₁₂ be the basin of attraction of E ₁₂ in ϖ. Then ϖ ₁₂ is open and forward invariant, and ϖ−ϖ ₁₂ is closed and forward invariant in ϖ. Suppose int(ϖ−ϖ ₁₂) is not empty. Since the boundary equilibria O and E ₁ are hyperbolic and cannot form a heteroclinic cycle, hypotheses of (H-1) to (H-4) derived by Butler et al. (1986) are satisfied in ϖ−ϖ ₁₂. Thus, system (1) restricted on ϖ−ϖ ₁₂ is uniformly persistent and has a positive equilibrium $E^{*}(N_{1}^{*},N_{2}^{*},N_{3}^{*})$ as a result of Butler et al. (1986). By the first two equations of (1), $E^{*}(N_{1}^{*},N_{2}^{*},N_{3}^{*})$ satisfies

  

  One the other hand, the equilibrium $P_{12}(N_{1}^{+},N_{2}^{+})$ of (2) satisfies

  

  It follows from the monotonicity of the functions (about N ₁) in the right-hand sides of the above equations that $N_{1}^{*} \le N_{1}^{+}$. By $N_{1}^{*} = N_{1}^{\#}$, we obtain $N_{1}^{\#}\le N_{1}^{+}$. Since

  $$-d_3 +\alpha_{31} N_1^+ /\bigl(c+ N_1^+\bigr)<0,\qquad -d_3 +\alpha_{31} N_1^\# /\bigl(c+ N_1^\#\bigr)=0 $$

  it follows from the monotonicity of function N ₁/(c+N ₁) that $N_{1}^{\#}> N_{1}^{+}$, which forms a contradiction. Thus, int(ϖ−ϖ ₁₂) is empty and (ii) is proved.

 □

Appendix D

Proof of Theorem 4.4

Since r ₁/d ₁<d ₂/(α ₂₁−ad ₂) and r ₁/d ₁>cd ₃/(α ₃₁−d ₃), we obtain $\mu_{2}^{(1)}<0$ and $\mu_{3}^{(1)}>0$. Then E ₁ is asymptotically stable on the (N ₁,N ₂)-plane. It follows from Theorem 3.2 that, system (2) either has no positive equilibrium, or has two positive equilibria.

Let N(t) be a solution of (1) with N _i (0)>0,i=1,2,3. If there is no positive equilibrium of (2) as shown in Theorem 3.2(i), E ₁ is globally asymptotically stable in the interior of the (N ₁,N ₂)-plane. Let $\bar{N}(t) = (\bar{N}_{1}(t),\bar{N}_{2}(t))$ be a solution of (2) with $\bar{N}_{i}(0) = N_{i}(0), i=1,2$, then $\lim_{t \to \infty} \bar{N}_{2}(t)=0$. By a proof similar to that of Theorem 4.3(ii), we obtain $N_{2}(t) \le \bar{N}_{2}(t)$ for t>0, which implies lim _t→∞ N ₂(t)=0.

For the case that both $P_{12}^{-}(N_{1}^{-},N_{2}^{-})$ and $P_{12}(N_{1}^{+},N_{2}^{+})$ are positive equilibria of (2) as shown in Theorem 3.2(iii), the stable manifold of equilibrium $P_{12}^{-}$ divides the (N ₁,N ₂)-plane into two regions. The region below the manifold, which is denoted by Ω ₁, is the basin of attraction of P ₁, while the other one, which is denoted by Ω ₁₂, is the basin of attraction of P ₁₂. Let $\bar{N}(t) = (\bar{N}_{1}(t),\bar{N}_{2}(t))$ be a solution of (2) with $(\bar{N}_{1}(0),\bar{N}_{2}(0)) \in \varOmega_{1}$, then $\lim_{t \to \infty} \bar{N}_{2}(t)=0$. By a proof similar to that of Theorem 4.3(ii), we obtain lim _t→∞ N ₂(t)=0 for solutions of (1) with (N ₁(0),N ₂(0))∈Ω ₁.

Let ϖ ₁⊆ϖ be a set such that solutions of (1) with N(0)∈ϖ satisfy lim _t→∞ N ₂(t)=0. Then ϖ ₁ is open and forward invariant, and ϖ−ϖ ₁ is closed and forward invariant in ϖ. It follows from $\mu_{2}^{(1)} <0$ and $N_{1}^{\#}<r_{1}/d_{1}$ that $\mu_{2}^{(13)}<0$. Similar to the proof of Theorem 4.2, we obtain $N_{1}^{\pm}> N_{1}^{\#}$ and $\mu_{3}^{(12)}>\mu_{3}^{(12-)}>0$. Then the boundary equilibria of (1) are hyperbolic and cannot form a heteroclinic cycle. By a proof similar to that of Theorem 4.2(iii), the set int(ϖ−ϖ ₁) is empty. That is, solutions of (1) with N ₂(0)>0 satisfy lim _t→∞ N ₂(t)=0. A similar proof can be given for the case in Theorem 3.2(ii). □

Appendix E

Proof of Theorem 4.5

It follows from r ₁/d ₁<d ₂/(α ₂₁−ad ₂) and r ₁/d ₁<cd ₃/(α ₃₁−d ₃) that $\mu_{i}^{(1)} < 0, i=2,3$. Then E ₁ is globally asymptotically stable in the interior of the (N ₁,N ₃)-plane by Theorem 3.4(i).

• (i) When Δ ₁<0 or a ₁≥0, it follows from Theorem 3.2(i) that equilibrium E ₁ is globally asymptotically stable in the interior of the (N ₁,N ₂)-plane. By a proof similar to that of Theorem 4.3(ii) and Theorem 4.4, we have lim _t→∞ N _i (t)=0,i=2,3.

• (ii) It follows from $\alpha_{31} < d_{3}(1 + c/N_{1}^{+})$ that $\mu_{3}^{(12)} < 0$. When Δ ₁>0 and a ₁<0, $E_{12}^{-}(N_{1}^{-},N_{2}^{-},0 )$ and $E_{12}(N_{1}^{+},N_{2}^{+},0 )$ are boundary equilibria of (1). Since function N ₁/(c+N ₁) is monotonic, we obtain $\mu_{3}^{(12-)} < 0$ by $\mu_{3}^{(12-)} < \mu_{3}^{(12)}$ and $\mu_{3}^{(12)} < 0$. Thus, E ₁₂ is asymptotically stable, while $E_{12}^{-}$ is a saddle point with a two-dimensional stable manifold $S_{12}^{-}$. It follows from Theorem 3.2(iii) and the comparison theorem that, every solution of (1) with N ₁(0)>0 satisfies either limsup _t→∞ N ₁(t)≤r ₁/d ₁,limsup _t→∞ N ₂(t)≤0, or $\lim \sup_{t \to \infty} N_{1}(t) \le N_{1}^{+}, \lim \sup_{t \to \infty} N_{2}(t) \le N_{2}^{+}$. For the first case, we have lim _t→∞ N ₂(t)=0 and then lim _t→∞ N ₃(t)=0. For the second case, by a proof similar to that of Theorem 4.3(ii), we obtain lim _t→∞ N ₃(t)=0. By Theorem 3.2(iii), $S_{12}^{-}$ divides int$R_{+}^{3}$ into two regions. One region, which is denoted by ϖ ₁, is the basin of attraction of E ₁. The other region, which is denoted by ϖ ₁₂, is that of E ₁₂. Since ϖ ₁ and ϖ ₁₂ are forward invariant, (ii) is proved.

• (iii) By a proof similar to that of (ii), ϖ ₁ is the basin of attraction of E ₁. In ϖ ₁₂, E ₁₂ is unstable in the N ₃-axis direction since $\mu_{3}^{(12)} > 0$. Thus, system (1) is uniformly persistent in ϖ ₁₂.

• (iv) Since function N ₁/(c+N ₁) is monotonic, we obtain $\mu_{3}^{(12)} >0$ by $\mu_{3}^{(12)} > \mu_{3}^{(12-)}$ and $\mu_{3}^{(12-)} > 0$. Then equilibria $E_{12}^{-}$ and E ₁₂ are saddle points. Since E ₁ is locally asymptotically stable, we denote the basin of attraction of E ₁ in ϖ by ϖ ₁. Then ϖ ₁ is open and forward invariant, and ϖ−ϖ ₁ is closed and forward invariant in ϖ. By a proof similar to that of Theorem 4.2(i), system (1) is uniformly persistent and has a positive equilibrium $E^{*}(N_{1}^{*},N_{2}^{*}, N_{3}^{*})$ in ϖ−ϖ ₁ as a result of Butler et al. (1986).

We denote the solution of (2) with $\bar{N}_{i}(0)=N_{i}^{*}$ by $\bar{N}(t), i=1,2$. By a proof similar to that of Theorem 4.3(ii), we have $\bar{N}_{i}(t) \ge N_{i}^{*}$ for t>0. It follows from $\mu_{3}^{(12-)} > 0$ that $-d_{3} + \alpha_{31} N_{1}^{-} /(c+N_{1}^{-})> -d_{3} + \alpha_{31} N_{1}^{*} /(c+N_{1}^{*}) = 0$. As shown in Fig. 1c, we denote the N ₁-isocline of (2) by l ₁:r ₁−d ₁ N ₁+α ₁₂ N ₂ f=0, and the N ₂-isocline of (2) by l ₂:−d ₂+α ₂₁ N ₁ f=0. The isocline l ₁ (resp. l ₂) and the N ₁-axis intersects at P ₁(r ₁/d ₁,0) (resp. Q ₁(d ₂/(α ₂₁−ad ₂),0)).

Since $\mu_{2}^{(1)}<0$, we have r ₁/d ₁<d ₂/(α ₂₁−ad ₂). Thus, Q ₁ is at the right-hand side of E ₁. Since there is no intersection of l ₁ and l ₂ as $N_{1} < N_{1}^{-}$, l ₂ is below l ₁ as $N_{1} < N_{1}^{-}$. Since $r_{1} - d_{1}N_{1} + \alpha_{12}N_{2}/(1+aN_{1}+bN_{2})|_{Q_{1}} <0$, the vector field of (2) satisfies dN ₁/dt<0 in the region below l ₁. Then l ₂ is below the stable manifold of $P_{12}^{-}$ when $N_{1} < N_{1}^{-}$, as shown in Fig. 1c. Since the point $(N_{1}^{*},N_{2}^{*})$ with $N_{1}^{*} < N_{1}^{-}$ is on l ₂, it is below the stable manifold. By Theorem 3.2(iii), we obtain $\lim_{t \to \infty} \bar{N}_{2}(t)=0$. That is, $N_{2}^{*}=0$, which forms a contradiction. □