Plant Dynamics, Birth-Jump Processes, and Sharp Traveling Waves

Abstract

Motivated by the importance of understanding the dynamics of the growth and dispersal of plants in various environments, we introduce and analyze a discrete agent-based model based on a birth-jump process, which exhibit wave-like solutions. To rigorously analyze these traveling wave phenomena, we derive the diffusion limit of the discrete model and prove the existence of traveling wave solutions (sharp and continuously differentiable) assuming a logarithmic-type growth. Furthermore, we provide a variational speed for the minimum speed of the waves and perform numerical experiments that confirm our results.

Appendices

A Center Manifold Theorem

Let $n\ge 2$ and consider the system

$$\begin{aligned} x' = F(x),\;x\in \mathbb {R}^n,\;F:\mathbb {R}^2\rightarrow \mathbb {R}^n, \end{aligned}$$

(39)

with $F\in C^r(\mathbb {R}^2)$ for $r\ge 2$ and fixed point $x_0$, i.e., that is $F(x_0) = 0$. Consider the linearized system of (39) about the

$$\begin{aligned} x' = Ax, \end{aligned}$$

where A is a $n\times n$ matrix. Assume that A has n real and distinct eigenvalues $\left\{ \lambda _1,\lambda _2,\ldots ,\lambda _n\right\}$ with corresponding eigenvectors $\left\{ v_1,v_2,\ldots , v_n\right\} .$

Definition 3

The set $E_s=span\left\{ v_i:\lambda _i<0\right\}$ is the stable subspace of the equilibrium $x_0$. The set $E_u=span\left\{ v_i:\lambda _i>0\right\}$ is the unstable subspace of the equilibrium $x_0$. The set $E_c=span\left\{ v_i:\lambda _i=0\right\}$ is the center subspace of the equilibrium $x_0$.

It is easily proved that $\mathbb {R}^n = E_s\oplus E_u\oplus E_c$. In the analysis to follow we will take advantage of the following theorem.

Theorem 3

(Center manifold theorem Bressan et al. 2003; Carr 1981) Let $f\in C^r$ be a vector field on $\mathbb {R}^n$ which vanishes at the origin, i.e., $f(0) = 0$ and let $A = Df(0)$. Let the stable, center, and unstable invariant subspaces associated with be as in Definition 3. Then, there exist $C^r$ stable and unstable manifolds $W_s, \;W_u$ tangent to $E_s,\;E_u$, respectively, and a $C^{r-1}$ center manifold $W_c$ tangent to $E_c$. Moreover, the manifolds $W_s,W_u$, and $W_c$ are invariant under the flow f.

Consider the case when the unstable manifold of our system is empty and the system can be written in the form:

$$\begin{aligned} \left\{ \begin{array}{l} x' = Ax + f(x,y)\\ y' = Bx + g(x,y), \end{array}\right. \end{aligned}$$

(40)

with $x\in \mathbb {R}^p$ and $y\in \mathbb {R}^q$ with $p+q=n$ where A has eigenvalues with zero real part and B has eigenvalues with negative real part. The center manifold can be represented as

$$\begin{aligned} W^c= \left\{ (x,y):y=\varphi (x)\right\} , \;\varphi (0)=D\varphi (0)=0,\;\varphi (x):U\rightarrow \mathbb {R}^q, \end{aligned}$$

with $U\subset \mathbb {R}^p$ containing the origin. A good approximation of the flow along $W^c$ is then given by:

$$\begin{aligned} x' = Ax + f(x,\varphi (x)). \end{aligned}$$

(41)

We will use the following reduction principle.

Theorem 4

(Reduction principle) If the origin of (4) is asymptotically stable (unstable) then the origin of (40) is also asymptotically stable (unstable).

In order to approximate $y=\varphi (x)$ we apply the chain rule on $\varphi (x)$ go back to the dynamics of (40). Indeed, we have

$$\begin{aligned} y' = D\varphi (x)x' = D\varphi (x)[Ax + f(x,\varphi )] = Bx + g(x,\varphi ). \end{aligned}$$

From this we obtain the expression on the manifold:

$$\begin{aligned} M[\varphi (x)] = D\varphi (x)[Ax + f(x,\varphi )] -Bx - g(x,\varphi )=0, \quad \varphi (0)=D\varphi (0)=0. \end{aligned}$$

(42)

We use the following result to find a suitable approximation of $\varphi (x).$

Theorem 5

(Approximation of $\varphi (x)$ ) If a function $\tilde{\varphi }(x)$ with $\tilde{\varphi }(0)=\tilde{D}\varphi (0)=0$ can be found such that $M[\tilde{\varphi }(x)] = O(\left| x\right| ^m)$ for $m>1$ as $\left| x\right| \rightarrow 0$ then it holds that

$$\begin{aligned} \varphi (x) =\tilde{\varphi }(x) + O(\left| x\right| ^m)\;\text {as}\;\left| x\right| \rightarrow 0. \end{aligned}$$

B Behavior About a Plane Non-Hyperbolic Point

In this section, we discuss the theory developed by Andronov et al. in Andronov et al. (1972) for the behavior of a point that is non-hyperbolic. For this purpose, consider the $2\times 2$ system

$$\begin{aligned} \left\{ \begin{array}{l} x' = ax+by + f(x,y)\\ y' = cx +dy+ g(x,y), \end{array}\right. \end{aligned}$$

(43)

where f, g are analytic functions around the origin with zero a unique isolated fixed point. Moreover, assume that

$$\begin{aligned} a+d =0\quad \text {and}\quad ad-bc=0. \end{aligned}$$

When $a=b=0$ (as is in our case) the change of variables $\tilde{x} = x$ and $\tilde{y} = (c/d)x+y$ changes (43) into a system of the form:

$$\begin{aligned} \left\{ \begin{array}{l} \tilde{x}' = \tilde{f}(\tilde{x},\tilde{y})\\ \tilde{y}' = \tilde{y} + \tilde{g}(\tilde{x},\tilde{y}), \end{array}\right. \end{aligned}$$

(44)

with $\tilde{f},\tilde{g}$ also analytic about the origin. First, we look for solutions to

$$\begin{aligned} \tilde{y} + \tilde{g}(\tilde{x},\tilde{y}) = 0, \end{aligned}$$

in a neighborhood of the origin. This solution $\varphi (\tilde{x})$ is obtained using the Implicit Function Theorem, which also guarantees that $\varphi (0)=\varphi '(0) =0$. Next, note that

$$\begin{aligned} \varPhi (\tilde{x}) := \tilde{f}(\tilde{x},\varphi (\tilde{x})), \end{aligned}$$

is not exactly zero since the origin is an isolated equilibrium. Hence, we expand $\varPhi (\tilde{x})$:

$$\begin{aligned} \varPhi (\tilde{x}) \approx K_m x^m +\cdots , \end{aligned}$$

with $m\ge 2$ and $K_m\ne 0.$ We will use the following theorem.

Theorem 6

(Andronov et al. 1972) Let (0, 0) be an isolated fixed point of (44), and let $\varphi (\tilde{x})$ and $\varPhi (\tilde{x})$ defined as above. Then:

• 1. If m is odd and $K_m>0$ then the origin is a topological node.

• 2. If m is odd and $K_m<0$ then the origin is a topological saddle point.

• 3. If m is even, then the origin is a saddle-node (it canonical neighborhood is the union of one parabolic and two hyperbolic sectors). If $K_m<0,$ the hyperbolic sector contains a segment of the positive x-axis bordering the origin and if $K_m>0$ they contain a segment of the negative x-axis.