Bifurcation Analysis of Models with Uncertain Function Specification: How Should We Proceed?

Abstract

When we investigate the bifurcation structure of models of natural phenomena, we usually assume that all model functions are mathematically specified and that the only existing uncertainty is with respect to the parameters of these functions. In this case, we can split the parameter space into domains corresponding to qualitatively similar dynamics, separated by bifurcation hypersurfaces. On the other hand, in the biological sciences, the exact shape of the model functions is often unknown, and only some qualitative properties of the functions can be specified: mathematically, we can consider that the unknown functions belong to a specific class of functions. However, the use of two different functions belonging to the same class can result in qualitatively different dynamical behaviour in the model and different types of bifurcation. In the literature, the conventional way to avoid such ambiguity is to narrow the class of unknown functions, which allows us to keep patterns of dynamical behaviour consistent for varying functions. The main shortcoming of this approach is that the restrictions on the model functions are often given by cumbersome expressions and are strictly model-dependent: biologically, they are meaningless. In this paper, we suggest a new framework (based on the ODE paradigm) which allows us to investigate deterministic biological models in which the mathematical formulation of some functions is unspecified except for some generic qualitative properties. We demonstrate that in such models, the conventional idea of revealing a concrete bifurcation structure becomes irrelevant: we can only describe bifurcations with a certain probability. We then propose a method to define the probability of a bifurcation taking place when there is uncertainty in the parameterisation in our model. As an illustrative example, we consider a generic predator–prey model where the use of different parameterisations of the logistic-type prey growth function can result in different dynamics in terms of the type of the Hopf bifurcation through which the coexistence equilibrium loses stability. Using this system, we demonstrate a framework for evaluating the probability of having a supercritical or subcritical Hopf bifurcation.

Appendices

Appendix 1: The First Lyapunov Exponent in a Planner System

The stability of a limit cycle is determined by its First Lyapunov number $L_{1}$. In the two-dimensional system

$$\begin{aligned} \dot{x}&= ax + by + f\left( {x,y} \right) ,\\ \dot{y}&= cx + dy + g\left( {x,y} \right) \end{aligned}$$

with $f\left( {x,y} \right) =a_{20} x^{2}+a_{11} xy+a_{02} y^{2}+a_{30} x^{3}+a_{21} x^{2}y+a_{12} xy^{2}+a_{03} y^{3}$, and $g\left( {x,y} \right) =b_{20} x^{2}+b_{11} xy+b_{02} y^{2}+b_{30} x^{3}+b_{21}x^{2}y+b_{12} xy^{2}+b_{03} y^{3}$, the first Lyapunov number is given by the following expression (Bautin and Leontovich 1976; Chow et al. 1994):

$$\begin{aligned} L_1&= -\frac{\pi }{4b\varDelta ^{\frac{3}{2}}}\left\{ \left[ ac\left( a_{11}^2 +a_{11} b_{02} +a_{02} b_{11} \right) +ab\left( b_{11}^2 +b_{11} a_{20} +b_{20} a_{11} \right) \right. \right. \\&\left. \left. +c^{2}\left( a_{11} a_{02} +2a_{02} b_{02} \right) -2ac\left( b_{02}^2 -a_{20} a_{02} \right) -2ab\left( a_{20}^2 -b_{20} b_{02} \right) \right. \right. \\&\left. \left. -b^{2}\left( 2a_{20} b_{20} +b_{11} b_{20} \right) +\left( bc-2a^{2} \right) \left( b_{11} b_{02} -a_{11} a_{20} \right) \right] \right. \\&\left. -\left( a^{2}+bc \right) \left[ 3\left( cb_{03} -ba_{30} \right) +2a\left( a_{21} +b_{12} \right) +\left( ca_{12} -bb_{21} \right) \right] \right\} , \end{aligned}$$

where $\varDelta$ is the determinant of the Jacobian matrix.

In the case where the first Lyapunov number, evaluated at the equilibrium at the Hopf bifurcation point, is positive, then the resulting limit cycle will be stable, and the Hopf bifurcation is supercritical. If the first Lyapunov exponent is negative, the resulting limit cycle will be unstable and so the Hopf bifurcation is a subcritical one (Kuznetsov 2004).

Appendix 2: Deriving the Projection from Function Space to the Space of Local Values

Here, we derive and sketch a proof of the necessary and sufficient conditions for the existence of a function $\tilde{r}$ in the $\varepsilon _Q$-neighbourhood of the base function $r$ taking the values $x^{*},\;\tilde{r} \left( {x^{*}} \right) ,\;\tilde{r} ^{{\prime }}\left( {x^{*}} \right) ,\;\tilde{r}^ {{\prime }{\prime }}(x^{*})$ and $\tilde{r}^ {{\prime }{\prime }{\prime }}(x^{*})$.

For $\tilde{r}$ to remain within the $\varepsilon _Q$-neighbourhood, it must first satisfy:

$$\begin{aligned} \tilde{r} \left( x \right) <r_{\varepsilon +} \left( x \right)&= 1+\varepsilon -x,\; \hbox {and}\\ \tilde{r} \left( x \right) >r_{\varepsilon -} \left( x \right)&= 1-\varepsilon -x. \end{aligned}$$

Essentially, this means that $\tilde{r} (x)$ must remain between the red bounds in Fig. 6 over the whole domain—it must remain within distance $\varepsilon$ of the base function.

$\tilde{r}$ must also be in $Q$, so must further satisfy:

• (i) $\tilde{r} ^{{\prime }{\prime }{\prime }}\left( x \right) =\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) \;\forall x\in \left( {x^{*}-w,x^{*}+w} \right) ,$

• (ii) $\left| {\tilde{r}^{{\prime }{\prime }}(x)} \right| <A \quad \forall x\in \left[ {0,x_{\hbox {max}} } \right] ,$

• (iii) $\tilde{r} ^{{\prime }}\left( x \right) <0\quad \forall x\in \left[ {0,x_{\hbox {max}} } \right] ,$

• (iv) $\tilde{r} \left( 0 \right) >0.$

Condition i) tells us that across the interval $\left( {x^{*}-w,x^{*}+w} \right) ,\; \tilde{r}$ is given by the cubic:

$$\begin{aligned} \tilde{r} \left( x \right) =\tilde{r} \left( {x^{*}} \right) +\tilde{r} ^{{\prime }}\left( {x^{*}} \right) \left( {x-x^{*}} \right) +\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) \left( {x-x^{*}} \right) ^{2}+\frac{\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) }{6}\left( {x-x^{*}}\right) ^{3} \end{aligned}$$

(9)

Therefore, an initial necessary condition for the existence of a valid function $\tilde{r}$ attaining the values $x^{*},\;\tilde{r} \left( {x^{*}} \right) ,\;\tilde{r} ^{{\prime }}\left( {x^{*}} \right) ,\;\tilde{r}^ {{\prime }{\prime }}(x^{*})$ and $\tilde{r}^ {{\prime }{\prime }{\prime }}(x^{*})$ is that this cubic must stay between $r_{\varepsilon -}$ and $r_{\varepsilon +}$ over this interval. In addition, this cubic must always have a negative first derivative and not have a second derivative of magnitude greater than $A$. Furthermore, $\tilde{r}$ will be bounded above by the parabolas tangent to the above cubic at $x^{*}-w$ and $x^{*}+w$ with second derivative $A$, and will be bounded below by the tangent parabolas with second derivative $-A$. These are given by the blue curves in Fig. 6. Taking these upper and lower bounds over the intervals $[0,x^{*}-w)$ and $(x^{*}+w,x_\mathrm{max}]$, together with the fact that $\tilde{r}$ must equal the cubic (B1) over $\left( {x^{*}-w,x^{*}+w} \right)$, we can construct the following functions:

$$\begin{aligned}&\!\!\!\!P_1 \left( x \right) \\&=\left\{ \begin{array}{ll} B+C\left( {x-x^{*}+w} \right) -\frac{A}{2}\left( {x-x^{*}+w} \right) ^{2}: &{} x\in [0,x^{*}-w) \\ \tilde{r} \left( {x^{*}} \right) +\tilde{r} ^{{\prime }}\left( {x^{*}} \right) \left( {x-x^{*}} \right) +\frac{\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) }{2}\left( {x-x^{*}} \right) ^{2}+\frac{\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) }{6}\left( {x-x^{*}} \right) ^{3}:&{} x\in \left( {x^{*}-w,x^{*}+w} \right) \\ D+E\left( {x-x^{*}-w} \right) -\frac{A}{2}\left( {x-x^{*}-w} \right) ^{2}: &{} x\in (x^{*}+w,x_\mathrm{max}]\\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned}&\!\!\!\! P_2 \left( x \right) \\&=\left\{ \begin{array}{ll} B+C\left( {x-x^{*}+w} \right) +\frac{A}{2}\left( {x-x^{*}+w} \right) ^{2}:&{}x\in [0,x^{*}-w) \\ \tilde{r} \left( {x^{*}} \right) +\tilde{r} ^{{\prime }}\left( {x^{*}} \right) \left( {x-x^{*}} \right) +\frac{\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) }{2}\left( {x-x^{*}} \right) ^{2}+\frac{\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) }{6}\left( {x-x^{*}} \right) ^{3}:&{}x\in \left( {x^{*}-w,x^{*}+w} \right) \\ D+E\left( {x-x^{*}-w} \right) +\frac{A}{2}\left( {x-x^{*}-w} \right) ^{2}:&{} x\in (x^{*}+w,x_\mathrm{max} ] \\ \end{array} \right. \end{aligned}$$

where:

$$\begin{aligned} B&= \tilde{r} \left( {x^{*}} \right) -w\tilde{r} ^{{\prime }}\left( {x^{*}} \right) +\frac{w^{2}}{2}\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) -w^{3}\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) ,\\ C&= \tilde{r} ^{{\prime }}\left( {x^{*}} \right) -w\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) +\frac{w^{2}}{2}\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) ,\\ D&= \tilde{r} \left( {x^{*}} \right) +w\tilde{r} ^{{\prime }}\left( {x^{*}} \right) +\frac{w^{2}}{2}\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) +w^{3}\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) ,\\ E&= \tilde{r} ^{{\prime }}\left( {x^{*}} \right) +w\tilde{r} ^{{\prime }{\prime }}\left( {x^{*}} \right) +\frac{w^{2}}{2}\tilde{r} ^{{\prime }{\prime }{\prime }}\left( {x^{*}} \right) . \end{aligned}$$

For any function $\tilde{r} \in Q, P_1$ and $P_2$ necessarily form lower and upper bounds for $\tilde{r}$, since they are constructed as the extreme cases of functions in $Q$. So we have $P_1 \left( x \right) \le \tilde{r}\left( x \right) \le P_2 \left( x \right) \;\forall x\in \left[ {0,x_\mathrm{max}} \right]$ (indeed, $\tilde{r} ,\; P_1$ and $P_2$ coincide over the interval $\left( {x^{*}-w,x^{*}+w} \right) )$, and therefore, the conditions:

$$\begin{aligned} P_1 \left( x \right)&< r_{\varepsilon -} \left( x \right) ,\\ P_2 \left( x \right)&< r_{\varepsilon +} \left( x \right) ,\\ P_1^{\prime } \left( x \right)&< 0\;\forall x\in \left( {x^{*}-w,x^{*}+w} \right) ,\\ \left| {P_1^{{\prime }{\prime }}(x)} \right|&< A\;\forall x\in \left( {x^{*}-w,x^{*}+w} \right) , \end{aligned}$$

are necessary (note: $P_1 ,\; P_2$ and $\tilde{r}$ coincide over the interval $\left( {x^{*}-w,x^{*}+w} \right)$ and so are interchangeable in the 3rd and 4th conditions). In terms of the figure, these conditions can be interpreted as requiring that the red upper and blue lower bounds clearly cannot cross.

It remains to be shown that they are sufficient. In order to prove this, it is enough to provide a method to construct a valid function $\tilde{r}$ which remains between $r_{\varepsilon -}$ and $r_{\varepsilon +}$ given only these conditions. We already have $\tilde{r}$ equal to $P_1$ and $P_2$ over $\left( {x^{*}-w,x^{*}+w} \right)$, so only need to construct $\tilde{r}$ over $[0,x^{*}-w)$ and $(x^{*}+w,x_{\hbox {max}} ]$. To do this, we can in fact use the exact same approach taken in Appendix 2 of Adamson and Morozov (2014).