Mathematical modeling of cascading migration in a tri-trophic food-chain system

Abstract

Diel vertical migration is a behavioral antipredator defense that is shaped by a trade-off between higher predation risk in surface waters and reduced growth in deeper waters. The strength of migration of zooplankton increases with a rise in the abundance of predators and their exudates (kairomone). Recent studies span multiple trophic levels, which lead to the concept of coupled vertical migration. The migrations that occur at one trophic level can affect the vertical migration of the next lower trophic level, and so on, throughout the food chain. This is called cascading migration. In this paper, we introduce cascading migration in a well-known model (Hastings and Powell, Ecology 73:896–903, 1991). We represent the dynamics of the system as proposed by Hastings and Powell as a phytoplankton–zooplankton–fish (prey–middle predator–top predator) model where fish affect the migrations of zooplankton, which in turn affect the migrations of motile phytoplankton. The system under cascading migration enhances system stability and population coexistence. It is also observed that for a higher rate of cascading migration, the system shows chaotic behavior. We conclude that the observations of Hastings and Powell remain true if the cascading migration rate is high enough.

Appendix

1.1 Appendix A: Proof of Theorem 1

Let us define a function

$$ \begin{array}{lll} W&=&x+y+z. \end{array} \label{EQ:eqn9} $$

(11)

The time derivative of Eq. (9) along with the solution of (8) is

$$ \begin{array}{rll} \frac{d W}{dt}&=&\frac{d x}{d t}+\frac{d y}{d t}+\frac{d z}{d t}=r\left(1-\frac{x}{\widetilde{K}}\right)-d_1 y-d_2 z\\ &\Rightarrow& \frac{dW}{dt}+\mu W=rx\left(1+\frac{\mu}{r}-\frac{x}{\widetilde{K}}\right)-(d_1-\mu)y-(d_2-\mu)z\\ &\leq& \frac{(r+\mu)^2\widetilde{K}}{4 r}=L ~(say), \end{array} $$

where μ ≤ min{d ₁,d ₂}.

Applying the theorem of differential inequality [52], we obtain $0 \leq W(x,y,z) \leq {L}/{\mu(1-e^{-\mu t})}+W(x_0,y_0,z_0)e^{-\mu t}$, which implies that 0 ≤ W ≤ L/μ as t → ∞. Hence, all the solutions of (8), that initiate in $R^3_+-\{0\},$ are confined in the region $B=\{(x,y,z)\in R^3_+: W=L/\mu+\epsilon \}$.

1.2 Appendix B: Proof of Theorem 2

For m = m ^*, we can write the characteristic equation

$$ \lambda^3+\Sigma_1 \lambda^2+\Sigma_2 \lambda+\Sigma_3=0 $$

as

$$ (\lambda^2+\Sigma_2)(\lambda+\Sigma_1), $$

which has three roots $\lambda_1=i\sqrt{\Sigma_2}$, $\lambda_2=-i\sqrt{\Sigma_2}$ and λ ₃ = − Σ₁.

For all m, the roots are in general of the form

$$ \begin{array}{rll} \lambda_1(m)&=&\phi_1(m)+i\phi_2(m),\\ \lambda_2(m)&=&\phi_1(m)-i\phi_2(m),\\ \lambda_3(m)&=&-\Sigma_1. \end{array} $$

Now, we shall verify the transversality condition

$$ \frac{d}{d m}(Re(\lambda(m)))\mid_{~m=m^*} \neq 0, ~~~j=1,2. $$

Substituting λ _j (m) = φ ₁(m) + iφ ₂(m) into the characteristic equation and calculating the derivative, we have

$$ \begin{array}{rll} P(m)\phi_1'(m)-Q(m)\phi_2'(m)+U(m)&=&0,\\ Q(m)\phi_1'(m)+P(m)\phi_2'(m)+V(m)&=&0, \end{array} $$

where

$$ \begin{array}{rll} P(m)&=&3 \phi_1^2(m)+2\Sigma_1(m)\phi_1(m)+\Sigma_2(m)-3 \phi_2^2(m),\\ Q(m)&=&6 \phi_1(m)\phi_2(m)+2 \Sigma_1(m)\phi_2(m),\\ U(m)&=&\phi_1^2(m)\Sigma_1'(m)+\Sigma_2'(m)\phi_1(m)+\Sigma_3'(m)-\Sigma_1'(m)\phi_2^2(m),\\ V(m)&=&2\phi_1(m)\phi_2(m)\Sigma_1'(m)+\Sigma_2'(m)\phi_2(m). \end{array} $$

Noticing that $\phi_1(m^*)=0$, $\phi_2(m^*)=\sqrt{\Sigma_2(m^*)}$, we have

$$ \begin{array}{rll} P(m^*)&=&-2\Sigma_2(m^*), Q(m^*)=2\Sigma_1(m^*) \sqrt{\Sigma_2(m^*)}, \\ U(m^*)&=&\Sigma_3'(m^*)-\Sigma_1'(m^*)\Sigma_2(m^*) \end{array} $$

and $V(m^*)=\Sigma_2'(m^*)\sqrt{\Sigma_2(m^*)}$. Now,

$$ \begin{array}{lll} &&{\kern-10pt}\dfrac{d}{d m}(Re(\lambda(m)))\mid_{~m=m^*} \\ &&= \dfrac{Q(m^*)V(m^*)+P(m^*)U(m^*)}{P(m^*)^2+Q(m^*)^2}\\ &&= \dfrac{2\Sigma_1(m^*) \sqrt{\Sigma_2(m^*)} \times \Sigma_2'(m^*)\sqrt{\Sigma_2(m^*)} + (-2\Sigma_2(m^*))(\Sigma_3'(m^*)-\Sigma_1'(m^*)\Sigma_2(m^*))}{(-2\Sigma_2(m^*))^2 + (2\Sigma_1(m^*) \sqrt{\Sigma_2(m^*)})^2}\\ &&=\dfrac{\Sigma_1(m^*)\Sigma_2'(m^*)-\Sigma_3'(m^*)+\Sigma_1'(m^*)\Sigma_2(m^*)}{2(\Sigma_2(m^*)+(\Sigma_1(m^*))^2)} \\ &&\neq 0, ~~if ~\Sigma_1(m^*)\Sigma_2'(m^*)-\Sigma_3'(m^*)+\Sigma_1'(m^*)\Sigma_2(m^*) \neq 0, \nonumber \end{array} \label{EQ:eqn9} $$

and $\lambda_3(m^*)=-\Sigma_1(m^*) \neq 0$.

Therefore, the transversality conditions hold. This implies that a Hopf bifurcation occurs at m = m ^*, hence the theorem.