Oscillations and Pattern Formation in a Slow–Fast Prey–Predator System

Abstract

We consider the properties of a slow–fast prey–predator system in time and space. We first argue that the simplicity of the prey–predator system is apparent rather than real and there are still many of its hidden properties that have been poorly studied or overlooked altogether. We further focus on the case where, in the slow–fast system, the prey growth is affected by a weak Allee effect. We first consider this system in the non-spatial case and make its comprehensive study using a variety of mathematical techniques. In particular, we show that the interplay between the Allee effect and the existence of multiple timescales may lead to a regime shift where small-amplitude oscillations in the population abundances abruptly change to large-amplitude oscillations. We then consider the spatially explicit slow–fast prey–predator system and reveal the effect of different timescales on the pattern formation. We show that a decrease in the timescale ratio may lead to another regime shift where the spatiotemporal pattern becomes spatially correlated, leading to large-amplitude oscillations in spatially average population densities and potential species extinction.

Appendices

Appendix A

Here, we will follow geometric singular perturbation technique as given by Fenichel (1979) to find the analytical expression of locally perturbed invariant manifold $C^1_{\varepsilon }$. Since $v=q(u,\varepsilon )$, from the invariance condition we have

$$\begin{aligned} \dfrac{\hbox {d}v}{\hbox {d}t} = \dfrac{\hbox {d}q(u,\varepsilon )}{\hbox {d}u}\dfrac{\hbox {d}u}{\hbox {d}t}. \end{aligned}$$

Using the explicit expression for $\dfrac{\hbox {d}u}{\hbox {d}t}$ and $\dfrac{\hbox {d}v}{\hbox {d}t}$ from (2), we get

$$\begin{aligned} \begin{aligned} \varepsilon q(u,\varepsilon )(u(1-\alpha \delta )-\delta ) = u \dfrac{\hbox {d}q(u,\varepsilon )}{\hbox {d}u}(\gamma (1-u)(u+\beta )(1+\alpha u)-q(u,\varepsilon )). \end{aligned} \end{aligned}$$

(35)

Substituting the asymptotic expansion of $q(u,\varepsilon )$ from (9 and assuming $u\ne 0$, $\dot{q_0}(u)\ne 0,$ we equate $\varepsilon$ free terms from both sides to obtain

$$\begin{aligned} \begin{aligned} q_0(u)&= \gamma (1-u)(u+\beta )(1+\alpha u),\\ \end{aligned} \end{aligned}$$

(36)

which is exactly the critical manifold. Now equating the coefficients of $\varepsilon$ from both sides of (35) we get

$$\begin{aligned} \begin{aligned} q_1(u) = \dfrac{q_0(u)(u(1-\alpha \delta )-\delta )}{-\,u{\dot{q}}_0(u)}. \end{aligned} \end{aligned}$$

(37)

Similarly, we obtain $q_2(u)$ by equating the coefficients of $\varepsilon ^2,$

$$\begin{aligned} \begin{aligned} q_2(u) = \dfrac{q_1(u)(u(1-\alpha \delta )-\delta )+u q_1\dot{q_1}(u)}{-\,u\dot{q_0}(u)}. \end{aligned} \end{aligned}$$

(38)

Proceeding as above, we find $q_r(u),\ r=3,4,\ldots$ by equating the coefficients of $\varepsilon ^r$ from (35). Therefore, the second-order approximation of the perturbed invariant manifold is given by

$$\begin{aligned} q(u,\varepsilon )=q_0(u)+\varepsilon q_1(u)+\varepsilon ^2 q_2(u), \end{aligned}$$

where $q_0,q_1,q_2$ are given in Eqs. (36)–(38).

Appendix B

We apply the blow-up transformation in the slow–fast normal form (12) where

$$\begin{aligned}&h_1(U,V)=u_*+U, \ \ h_3(U,V)=0,\ \ h_5(U,V)=(v_*+V)(1+\alpha u_*)+Uv_*\alpha , \\&h_2(U,V)=-\,\gamma (-1+6u_*^2\alpha +3u_*(1+\alpha (\beta -1))+\beta -\alpha \beta )-U\gamma (1+\alpha (4u_*+\beta -1)), \\&h_4(U,V)=(v_*+V)(1-\alpha \delta _*),\ \ h_6(U,V)=u_*-(1+u_*\alpha )\delta _*, \end{aligned}$$

${\bar{\varepsilon }}=1$ the blow-up transformation as defined in (13) reduces to

$$\begin{aligned} {\bar{r}} = \sqrt{\varepsilon },\ U=\sqrt{\varepsilon }{\bar{U}},\ V=\varepsilon {\bar{V}},\ \lambda =\sqrt{\varepsilon }{\bar{\lambda }}. \end{aligned}$$

(39)

Using the transformation (39), we can write the system (15) by removing the overbars as

$$\begin{aligned} \begin{aligned} U_t&= -\,b_1V+b_2U^2+\sqrt{\varepsilon }{\mathcal {G}}_1(U,V) + O(\sqrt{\varepsilon }(\lambda +\sqrt{\varepsilon })),\\ V_t&= b_3U-b_4\lambda +\sqrt{\varepsilon }{\mathcal {G}}_2(U,V) + O(\sqrt{\varepsilon }(\lambda +\sqrt{\varepsilon })),\\ \end{aligned} \end{aligned}$$

(40)

where

$$\begin{aligned} \begin{aligned} \&b_1 =u_*,\ \ b_2 =-\,\gamma (-1+6u_*^2\alpha +3u_*(1+\alpha (\beta -1))+\beta -\alpha \beta ),\\ \&b_3 = v_*(1-\alpha \delta _*),\ \ b_4 = v_*(1+\alpha u_*), \end{aligned} \end{aligned}$$

(41)

and

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_1(U,V) = a_1U-a_2UV+a_3U^3,\ \ {\mathcal {G}}_2(u,V) = a_4U^2+a_5V. \end{aligned} \end{aligned}$$

(42)

Let the equilibrium point of the system (40) is $(U_e,V_e)$, $U_e=\dfrac{b_4\lambda }{b_3}+O(2)$ and $V_e=O(2)$ where $O(2):=O(\lambda ^2,\lambda \sqrt{\varepsilon },\lambda )$. Linearizing the system about this equilibrium point we have the Jacobian matrix as

$$\begin{aligned} {\mathcal {J}}:= \begin{pmatrix} 2U_eb_2+a_1 \sqrt{\varepsilon }+O(2)&{}-b_1+O(2)\\ b_3+O(2)&{}a_5\sqrt{\varepsilon }+O(2) \end{pmatrix}. \end{aligned}$$

(43)

At the Hopf bifurcation, we have Trace ${\mathcal {J}}=0$ which implies

$$\begin{aligned} \dfrac{2b_2b_4\lambda }{b_3}+\sqrt{\varepsilon }(a_1+a_5)+O(2)=0. \end{aligned}$$

(44)

and applying the blow-down map ${\lambda _\mathrm{H}}=\lambda \sqrt{\varepsilon }$ we get the singular Hopf bifurcation curve ${\lambda _\mathrm{H}}(\sqrt{\varepsilon })$ for the slow–fast normal form (12) as

$$\begin{aligned} {\lambda _\mathrm{H}}(\sqrt{\varepsilon })=-\,\dfrac{b_3(a_1+a_5)}{2b_2b_4}\varepsilon +O(\varepsilon ^{3/2}). \end{aligned}$$

(45)

Appendix C

Here, we prove the existence of maximal canard curve and will give an analytical expression for the same. For that we will first prove the following proposition. In chart $K_2$ of the blow-up space, we consider the desingularized system (40) as

$$\begin{aligned} \begin{aligned} U_t&= -\,b_1V+b_2U^2+r{\mathcal {G}}_1(U,V) + O(\lambda r,r),\\ V_t&= b_3U-b_4\lambda +r{\mathcal {G}}_2(U,V) + O(\lambda r,r),\\ r_t&=0,\\ \lambda _t&=0, \end{aligned} \end{aligned}$$

(46)

where $b_1,\ b_2,\ b_3,\ b_4, {\mathcal {G}}_1$ and ${\mathcal {G}}_2$ are computed above (41), (42). The dynamics of the system on the sphere is obtained by putting $r=0$ in (46) for different values of $\lambda$ in the vicinity of 0. Thus, by taking $r=0,\ \lambda =0$, the above system is integrable and we have

$$\begin{aligned} \begin{aligned} U_t&= -\,b_1V+b_2U^2,\\ V_t&= b_3U. \end{aligned} \end{aligned}$$

(47)

This is a Riccati equation and the solution of this equation helps in proving our main theorem.

Proposition 1

The solution of the system (47) is given by $H(U,V) = c,$ where

$$\begin{aligned} H(U,V) = e^{-\dfrac{2b_2}{b_3}V}\left( \dfrac{b_3}{2}U^2-\dfrac{b_1b_3^2}{4b_2^2}-\dfrac{b_1b_3}{2b_2}V\right) \end{aligned}$$

and

$$\begin{aligned}&\dfrac{\hbox {d}U}{\hbox {d}t} = -\,e^{\dfrac{2b_2}{b_3}V}\dfrac{\partial H}{\partial V}, \nonumber \\&\dfrac{\hbox {d}V}{\hbox {d}t} = e^{\dfrac{2b_2}{b_3}V}\dfrac{\partial H}{\partial U}. \end{aligned}$$

(48)

Proof

We can write the above Riccati system (47) as

$$\begin{aligned} \dfrac{\hbox {d}V}{\hbox {d}U}=\dfrac{b_3U}{-b_1V+b_2U^2} \end{aligned}$$

(49)

where the integrating factor is $e^{-\dfrac{2b_2}{b_3}V}$. Multiplying both sides with the I.F and integrating we get

$$\begin{aligned} e^{-\dfrac{2b_2}{b_3}V} \left( U^2-\dfrac{b_1}{b_2}V-\dfrac{b_1b_3}{2b_2^2}\right) =c_0. \end{aligned}$$

Multiplying with $\dfrac{b_3}{2}$, we obtain the solution of the system( 47) as

$$\begin{aligned} e^{-\dfrac{2b_2}{b_3}V}\left( \dfrac{b_3}{2}U^2-\dfrac{b_1b_3^2}{4b_2^2}-\dfrac{b_1b_3}{2b_2}V\right) = c, \end{aligned}$$

where $c=c_0\dfrac{b_3}{2}$ is a constant. The solution determined by $c=0$ is a parabola of the form

$$\begin{aligned} U^2=\dfrac{b_1b_3}{2b_2^2}+\dfrac{b_1}{b_2}V. \end{aligned}$$

$\square$

Proof of theorem 4.2:

We write the solution of the system (47) in the parametric form

$$\begin{aligned} \eta (t) = (U(t),V(t))= \left( t,\dfrac{b_2}{b_1}t^2-\dfrac{b_3}{2b_2}\right) ,\ t\in \mathbb {R} \end{aligned}$$

(50)

For $\varepsilon =0$, the attracting and repelling submanifolds of the critical manifold ${\mathcal {M}}^1_0$ intersect along the equator of the blow-up space $S^3$. From Fenichel’s theory, for $\varepsilon >0$ there exist invariant perturbed attracting $({\mathcal {M}}_{\varepsilon }^{1,a})$ and repelling submanifold $({\mathcal {M}}_{\varepsilon }^{1,r})$. Along the curve (50), the attracting $({\mathcal {M}}_{\varepsilon }^{1,a})$ and repelling $({\mathcal {M}}_{\varepsilon }^{1,r})$ invariant submanifolds in the blow-up space intersect and the solution trajectory lying in that intersection is called maximal canard. We use Melnikov function to calculate the distance between these invariant manifolds (Krupa and Szmolyan 2001a; Kuehn 2015), which is given by

$$\begin{aligned} D_{r,\lambda } = d_r r + d_{\lambda } \lambda + O(r^2), \end{aligned}$$

(51)

where

$$\begin{aligned} \begin{aligned} d_r = \int _{-\infty }^{\infty }\nabla H(\eta (t))^T{\mathcal {G}}(\eta (t))\hbox {d}t,\\ d_{\lambda } = \int _{-\infty }^{\infty }\nabla H(\eta (t))^T\begin{pmatrix} 0\\ -b_4 \end{pmatrix}\hbox {d}t, \end{aligned} \end{aligned}$$

(52)

where ${\mathcal {G}},\ H$ and $b_4$ are defined in (42), (48) and (41), respectively. The distance between the submanifolds ${\mathcal {M}}_{\varepsilon }^{1,a}$ and ${\mathcal {M}}_{\varepsilon }^{1,r}$ is given by Eq. (51). And since the maximal canard lie in the intersection of these manifolds, so we must have $D_{r,\lambda }=0$. For that, we now calculate the Melnikov-type integrals $d_r$ and $d_{\lambda }$ (50 ) and (52). Therefore,

$$\begin{aligned} \begin{aligned} d_r&= \int _{-\infty }^{\infty }\left[ (a_1U-a_2UV+a_3U^3 )\dfrac{\partial H(\eta (t))}{\partial U}+(a_4U^2+a_5V)\dfrac{\partial H(\eta (t))}{\partial V}\right] \hbox {d}t\\&= \int _{-\infty }^{\infty }e^{-\dfrac{2b_2}{b_3}V}\left[ (a_1U-a_2UV+a_3U^3 )b_3U+(a_4U^2+a_5V)(b_1V-b_2U^2)\right] \hbox {d}t\\&= e\int _{-\infty }^{\infty }e^{-A_4t^2}\left( A_1t^4+A_2t^2+A_3\right) \hbox {d}t \end{aligned} \end{aligned}$$

(53)

where

$$\begin{aligned} A_1 = a_3b_3-\dfrac{a_2b_2b_3}{b_1},\ A_2 = a_1b_3+ \dfrac{a_2b_3^2}{2b_2}-\dfrac{a_4b_1b_3}{2b_2}-\dfrac{a_5b_3}{2},\ A_3 = \dfrac{a_5b_1b_3^2}{4b_2^2},\ A_4 = \dfrac{2b_2^2}{b_1b_3}. \end{aligned}$$

Now substituting $z=t^2$ and by repeated integration by parts, we obtain

$$\begin{aligned} \begin{aligned} d_r = e\left( \dfrac{3A_1}{4A_4^2}+\dfrac{A_2}{2A_4}+A_3\right) \int _{-\infty }^{\infty }e^{-A_4t^2}\hbox {d}t, \end{aligned} \end{aligned}$$

(54)

and

$$\begin{aligned} \begin{aligned} d_{\lambda }&= -\,\int _{-\infty }^{\infty }b_4\dfrac{\partial H}{\partial V}\hbox {d}t\\&= b_4\int _{-\infty }^{\infty }e^{-\dfrac{2b_2}{b_3}V}(-b_1V+b_2U^2)\hbox {d}t\\&= e A_5\int _{-\infty }^{\infty }e^{-A_4t^2}\hbox {d}t, \end{aligned} \end{aligned}$$

(55)

where $A_5 = \dfrac{b_1b_3b_4}{2b_2}$. Since $d_{\lambda }\ne 0$ therefore using implicit function theorem we can explicitly solve for $\lambda$ from (51)

$$\begin{aligned} \begin{aligned} \lambda (r)&=-\,\dfrac{\hbox {d}_r}{\hbox {d}_\lambda }r + O(r^2) = -\dfrac{1}{A_5}\left( \dfrac{3A_1}{4A_4^2}+\dfrac{A_2}{2A_4}+A_3\right) r + O(r^2). \end{aligned} \end{aligned}$$

(56)

Now using blow down map $\lambda _c=\lambda \sqrt{\varepsilon }$, we obtain the maximal canard curve for the slow–fast normal form (12).

$$\begin{aligned} \begin{aligned} \lambda _c(\sqrt{\varepsilon }) = -\,\dfrac{1}{A_5}\left( \dfrac{3A_1}{4A_4^2}+\dfrac{A_2}{2A_4}+A_3\right) \varepsilon + O(\varepsilon ^{3/2}). \end{aligned} \end{aligned}$$

(57)

$\square$

Appendix D

Here, we prove the existence of a unique attracting limit cycle called relaxation oscillation. To study the dynamics of the system (21), we define two sections of the flow as

$$\begin{aligned} \begin{aligned}&\varDelta ^\mathrm{in}=\{(u_+,v):u_+<<u_\mathrm{max}, v\in (v_1-\rho ,v_1+\rho )\},\\&\varDelta ^\mathrm{out}=\{(u_+,v):u_+<<u_\mathrm{max}, v\in (v_0-\rho ^2,v_0+\rho ^2)\}, \end{aligned} \end{aligned}$$

where $u_\mathrm{max},\ v_1,\ v_0$ are defined in Sect. 4.3 and $\rho$ is sufficiently small positive number.

Let us define a return map $\varPi :\varDelta ^\mathrm{in}\rightarrow \varDelta ^\mathrm{in}$ which is a composition of two maps

$$\begin{aligned} \varPhi :\varDelta ^\mathrm{in}\rightarrow \varDelta ^\mathrm{out},\ \ \varPsi :\varDelta ^\mathrm{out}\rightarrow \varDelta ^\mathrm{in}, \end{aligned}$$

such that $\varPi = \varPsi \circ \varPhi$. Let us fix $\varepsilon >0$ and we take a point $(u_+,v_+)$ on the section $\varDelta ^\mathrm{in}$. Now we consider a trajectory of the system (21) starting from the initial point $(u_+,v_+)$. From the analysis of the entry–exit function, we can say that this trajectory will be attracted to $V_+$ and will leave $V_-$ at point $(0,p(v_+)),$ where p is the entry–exit function. The trajectory then jumps into the section $\varDelta ^\mathrm{out}$ at the point $(u_+,p(v_+)).$ Thus, the map $\varPhi$ is defined with the help of entry–exit function as $\varPhi (u_+,v_+)=(u_+,p(v_+)).$

Now to study the map $\varPsi$ we consider two trajectories $\gamma _{\varepsilon }^1, \gamma _{\varepsilon }^2$ starting from the section $\varDelta ^\mathrm{out}$. These trajectories get attracted toward $C_{\varepsilon }^{1,a}$ where the slow flow is given by $\dfrac{\hbox {d}u}{\hbox {d}\tau }=\dfrac{g(u,q(u,\varepsilon ))}{{\dot{q}}(u,\varepsilon )}.$

They follow the slow perturbed manifold until the vicinity of the fold point where they contract exponentially toward each other (Wang and Zhang 2019b) and jump into $\varDelta ^{in}.$ From Theorem 2.1 of Krupa and Szmolyan (2001a), we have that the map $\varPi$ is a contraction. Using contraction mapping theorem, we conclude that $\varPi$ has a unique fixed point which gives rise to a unique relaxation oscillation cycle $\gamma _{\varepsilon }$. Further from Fenichel’s theory, we infer that $\gamma _{\varepsilon }$ converges to $\gamma _0$ as $\varepsilon \rightarrow 0.$

For the parameter values $\alpha =0.5,\ \beta =0.2,\ \delta =0.3,$ the unique attracting cycle $\gamma _{\varepsilon }$ for $\varepsilon =0.1$, is shown in Fig. 15 which converges to $\gamma _0$ as $\varepsilon \rightarrow 0$.

Fig. 15

Singular trajectory $\gamma _0$ (blue) and unique attracting limit cycle $\gamma _{\varepsilon }$ for $\varepsilon =0.1$ (green) for $\alpha =0.5,\ \beta =0.2,\ \delta =0.3$ (Color figure online)