System of quadratic autonomous ODEs - convexity of the solution curve

Question

_{Crossposted on MathOverflow}

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Problem:

For a given parameter $a>0$, consider the following autonomous system of ODEs for $(x,y,z): \mathbb R_+\to [0,1)^3$: \begin{align*} \dot{x}_t &= (1-x_t) (z_t-x_ty_t) &=:F^x(x_t,y_t,z_t) \\ \dot{y}_t &= \tfrac 1 2 y_t^2 - (a+x_ty_t)(1-y_t) &=:F^y(x_t,y_t,z_t) \\ \dot{z}_t &= \tfrac 1 2 z_t^2 - \tfrac 1 2 y_t^2 + (a+x_ty_t)z_t &=:F^z(x_t,y_t,z_t) \end{align*} together with the initial condition $x_0=0$. By solution I refer to the profile $(x_t,y_t,z_t)\in [0,1)^3$ that is defined for all $t\geq 0$ (there is unique point $(0,y_0,z_0)\in [0,1)^3$ such that the orbit passing through it does not explode).
Prove (analytically) that a) the curve consisting of points $(x_t,y_t)_{t\geq 0}$ is a convex; b) the curve consisting of points $(x_t,z_t)_{t\geq 0}$ is decreasing.

Question: Reference to any related paper / book.

Figure 1: Solution as a function of $t$ (for $a=0.1$). $t$" />

Figure 2: Solution represented by functions $Y,Z$ such that $(x_t,y_t,z_t)= (x_t,Y(x_t),Z(x_t))$ for all $t\geq 0$ (the red point represents the critical point of the system):

$Y$ and $Z$." />

Note: I called the system quadratic in the title because substituting $u_t = x_t y_t$ converts the original (cubic) system into a quadratic one.

Context: I obtained this system of ODEs when studying patent race with private information: Two firms compete making a patent, they exert a costly effort that translates into the hazard rate of making a discovery. To patent a firm has to make two consecutive discoveries, the first one that does so wins (reward 1) whilst the other loses (reward 0). Having made the first discovery is firms private information, the rival only infers a posterior belief about it based on the fact that nobody patented yet. In the presented equations this belief is represented by variable $x_t$; $y_t$ is the effort of a firm that has made the first discovery already; $z_t$ is the effort of a firm that has made no discovery yet, and the parameter $a>0$ is the rate at which future payoffs are being discounted.

Basic Observations

I have analytical proofs of the following properties:

• The system has a unique critical point $(x_*,y_*,z_*)\in [0,1)^3$.
• For every solution, $t\mapsto x_t$ is increasing and the trajectory $(x_t,y_t,z_t)$ converges to the critical point $(x_*,y_*,z_*)$.
• The Jacobian $J$ of the system has one negative eigenvalue $\lambda_1$ and the other eigenvalues have strictly positive real parts. Thus, by Hartman-Grobman Theorem the eigenvector $v=(v^x,v^y,v^z)$ associated with $\lambda_1$ determines the direction in which a trajectory can converge to the critical point, and there is unique local solution near the critical point with $x_t$ increasing.
• The local solution can be extended till $x=0$ is reaches, so there exist unique functions $Y(x),Z(x):[0,x_*)\to [0,1)$ such that $y_t = Y(x_t)$ and $z_t = Z(x_t)$ for any solution of the initial problem with the initial condition $x_0=\hat x$.

Current state of research:

I found a way to prove some basic properties like that $Y(x)$ is increasing and $Z(x)$ is decreasing and I'm working on proving that $Y(x)$ is convex. However, I prove everything using methods that I develop using just elementary mathematical analysis. I belive there should be some standard methods that I could apply instead of developing my own.

More details on the methods that I use:

The method that I've been using utilises the fact that at a given $\alpha>0$ and $x\in [0,x^*]$ $$y\mapsto G(y;x,\alpha)=F^y(x,y,Z(x)) - \alpha F^x(x,y,Z(x))$$ is a quadratic function and it can be shown that $Y'(x) \geq \alpha$ whenever $G(y;x,\alpha)>0$. Then I consider the isocline $\hat Y(x;\alpha)$ given implicitly by $G(\hat Y(x;\alpha);x,\alpha)=0$, i.e. the value that $y$ would need to have at $x$ in order for the slope $Y'(x;y)$ of the orbit passing by $(x,y,Z(x))$ to be equal to $\alpha$. In the next step I show that $\hat Y'(x;\alpha) < \alpha$ and so the curves $Y(x)$ and $\hat Y(x;\alpha)$ only cross at the critical point $x_*$. This way I can show some desirable properties of $Y(x)$, using some assumptions on $Z(x)$. Then I use similar method to prove properties of $Z(x)$ using assumptions on $Y(x)$. Finally, I show that both properties of $Y(x)$ and $Z(x)$ hold (without the argument being cyclical).

I tried to formulate an abstraction / simplification of this question:

• Convexity / concavity of $(g_t,x_t)$, where $x_t$ solves the ODE $\dot x_t=\tfrac 1 2 x_t^2 - (1-x_t) g_t$.

• Solution $x:\mathbb R_+\to [0,1]$ of the ODE $\dot x = \tfrac 1 2 x^2 - (1-x) (1-e^{-at})$ is a concave function of $g(t)=1-e^{-at}$.

Objective: I'd be grateful for any reference to a related studies.