Growth of a tree with allocations rules: Part 1 Kinematics

TL;DR

This paper introduces a non-local model for the growth of tree-like transportation networks, analyzing how resource allocation rules influence network viability, with analytical and numerical insights into different growth outcomes.

Contribution

It presents a novel non-local model for tree growth based on resource transport and allocation rules, with analytical and numerical analysis of network viability.

Findings

Viability depends on resource allocation and maintenance costs.

Different growth outcomes are identified through analytical and numerical methods.

Model extensions are discussed for future research.

Abstract

A non-local model describing the growth of a tree-like transportation network with given allocation rules is proposed. In this model we focus on tree like networks, and the network transports the very resource it needs to build itself. Some general results are given on the viability tree-like networks that produce an amount of resource based on its amount of leaves while having a maintenance cost for each node. Some analytical studies and numerical surveys of the model in "simple" situations are made. The different outcomes are discussed and possible extensions of the model are then discussed.

Figures

Equations

$$\mathcal{P}=\sum _{h=0}^H{E}_h\times p_0$$

$$\mathcal{M}=\sum _{h=0}^H(h+1)\times E_h\times m_0$$

$$\mathcal{B}=\mathcal{P}-\mathcal{M}=\sum _{h=0}^H{E}_h(p_0-(h+1)m_0)$$

$$H_0=\left[\frac{p_0}{m_0}-\frac{N_{max}}{N_{max}-1}\right]$$

$$\begin{array}[]{ccc}\mathcal{M}&=&m_{0}(27^{\alpha}+3\times 9^{\alpha}+9\times 3^{\alpha}+27\times 1^{\alpha})\\ &=&m_{0}{\displaystyle\sum_{i=0}^{3}}3^{i}\times 3^{(3-i)\alpha}\\ \mathcal{M}&=&3^{3\alpha}m_{0}{\displaystyle\sum_{i=0}^{3}}3^{(1-\alpha)i}\end{array}$$

$$\begin{array}[]{cccc}\mathcal{M}_{H}&=&N^{\alpha H}m_{0}{\displaystyle\sum_{i=0}^{H}}N^{(1-\alpha)i}\\ \\ &=&N^{H}m_{0}\frac{N^{1-\alpha}-N^{-(1-\alpha)H}}{N^{1-\alpha}-1}&\mathrm{,\>if\>}N>1\mathrm{\>and\>\alpha\neq 1}\end{array}$$

$${\mathcal{B}}_H=N^H\left(p_0-m_0\frac{N^{1-\alpha }-N^{-(1-\alpha )H}}{N^{1-\alpha }-1}\right)$$

$$\begin{array}[]{c}p_{critical}=\frac{N^{1-\alpha}}{N^{1-\alpha}-1}\\ p_{0}/m_{0}\geq p_{critical}\textrm{ }\Rightarrow\forall H,\textrm{ }\mathcal{B}_{H}\geq 0\\ p_{0}/m_{0}<p_{critical}\textrm{ }\Rightarrow\exists H_{f},\textrm{ }\forall H\geq H_{f},\textrm{ }\mathcal{B}_{H}<0\end{array}$$

$$\begin{array}[]{ccc}\mathcal{C}_{vol(H-1\rightarrow H)}&=&C_{v}\textrm{(number of additional extremities)}H\\ &=&N^{H}C_{v}(1-N^{-1})H\end{array}$$

$$\begin{array}[]{c}u_{H+1}=\left[\frac{B_{H+1}\left({\displaystyle p_{0}-\sum_{i=0}^{H+1}\left(\frac{B_{i}}{B_{H+1}}\right)^{1-\alpha}m_{0}}\right)+R_{H+1}}{\mathcal{C}B_{H+1}}\right]\end{array}$$

$$R_{H+1}=B_H\left(p_0-\sum _{i=0}^H{\left(\frac{B_i}{B_H}\right)}^{1-\alpha }{m}_0\right)+R_H-B_{H+1}\mathcal{C}$$

$$B_{H+1}=B_H\cdot \mathbf{min}\left(\left[\frac{p_0-{\sum }_{i=0}^H{\left(\frac{B_i}{B_H}\right)}^{1-\alpha }{m}_0}{\mathcal{C}}\right],N_{max}\right)$$

$$W_{n+1}={\left(\frac{B_{n\text{+1}}}{B_{n+1}}\right)}^{1-\alpha }+\sum _{i=0}^n{\left(\frac{B_i}{B_{n+1}}\right)}^{1-\alpha }=1+\sum _{i=0}^n{\left(\frac{B_i}{B_{n+1}}\right)}^{1-\alpha }.$$

$$\begin{array}[]{c}W_{n+1}=1+W_{n}\dfrac{1}{\left(\mathbf{min}\left(\left\lfloor a_{0}-W_{n}\cdot b_{0}\right\rfloor\>,\>N_{max}\right)\right)^{1-\alpha}}\\ \\ \textrm{Where }a_{0}=p_{0}/\mathcal{C}\textrm{ and }b_{0}=m_{0}/\mathcal{C}\\ \\ \textrm{with }W_{0}=1\end{array}$$

$$u_n=\mathbf{min}\left(\left[a_0-W_n{b}_0\right],N_{max}\right),$$

$$\begin{array}[]{ccc}(f_{1}-\mathbb{I})^{\prime}(x)&=&\dfrac{(1-\alpha)b_{0}x+a_{0}-b_{0}x-(a_{0}-b_{0}x)^{2-\alpha}}{(a_{0}-b_{0}x)^{2-\alpha}}\\ &=&\dfrac{a_{0}(1-\alpha)+\alpha(a_{0}-b_{0}x)-(a_{0}-b_{0}x)^{2-\alpha}}{(a_{0}-b_{0}x)^{2-\alpha}}\end{array}\>.$$

$$P^{\prime }(x)=b_0(-\alpha +(2-\alpha )(a_0-b_{0}x{)}^{1-\alpha })$$

Full text

1 Introduction

Systems of transportation frequently appear in physics, engineering, biology: transportation of water, electricity and gas in cities, street network, river basinsrodrig or vascular systemsBohn2002 ; PhysRevE.73.061907 etc. A typical way to formalize systems of transportation is to transform it in a problem of flow in a network between sources and sinksford62flows ; aaarticle ; ezvereefkpor with discussions about finding the best path from sources to sinks. Such approach can lead to application in economic contextFulkerson59 ; Fulkerson1977 ; Hougaard2009 . More physical approaches involving transport of fluids or electricity have been considered too and, in these cases, use of the formalism of electrical circuit (Kirchoff’s law, current, potential etc.) is not unusualPhysRevLett.98.088702 ; PhysRevLett.104.048703 , the more geometrical aspects would also be involvedPhysRevE.73.016116 ; PhysRevLett.98.088701 . This approach is often focused one minimizing some dissipative energy which gives it some application in engineering context but also in the description of nature as natural networks may arise from minimization of some dissipationRodriguez-Iturbe92 ; Banavar2001 .

We can notice that the focus is typically on optimization. The substance being transported is ether not a resource to be used or consumed or its use is of little importance to the problem the papers are studying. A possible problem with such approach is that real life networks may not just form from minimization/maximization problems but from some interaction between to flow and the network: vascular system of plants is a network transporting sugars, but these sugars are also used to build the network itself. Perennial plants (including trees) do not rely on a centralized organs such as a heart or a brain to grow or distribute the resource throughout the plants and yet how they branch and their final shape is not totally planned at birth either but is also strongly caused by adaptation to external stimuliChehab2008 . Given this, we may believe trees are an example of self-organization arising from very simple local rules and interactionsSACHS2004197 . The vascular system of trees is composed of a part called xylem and another one called the phloem. The xylem is mainly responsible for water transportation. This flow of water is unidirectional: it goes from the roots to the leaves then most of it gets evaporated at the level of leaves. These leaves then play a major role for the growth and sustenance of the organism: they are the ones absorbing carbon in the air and create sugars through photosynthesis. These sugars are essential as they sustain the living cells of the tree as well as provide its building materials. The part of the vascular system responsible for transporting this vital resource is the phloem which is the part we are interested with since it is the one that potentially showcases this idea of interaction between the network and the resource being transported. The biophysics of the phloem has been extensively studiedmunch_stoffbewegungen_1930 ; DeSchepper2013 ; RevModPhys.88.035007 .

However, instead of studying the biomechanics of sugar transport like the aforementioned papers, our approach is more abstract and in more line with the work about flow, sources and sinks mentioned earlier but we also propose to consider as well how the flow can be used to built the network itself while in turn the flow depends on the network and make a simple dynamical system out of it. In order to start on relatively simple grounds the toy-model we built and present in this paper will only use tree network (i.e. loopless). This has both the advantage of keeping the network topology quite tractable, while potentially shed some light on actual possible growth mechanisms occurring in biological trees. The dynamical and growing aspects we added also makes it similar to cellular automata which was a concept introduced by John Von Neumann in von1996theory , though a more famous example of cellular automaton is the Game of Life invented by John Conwaygardner1970mathematical .

In short, we are studying growing systems for which the growth and self-organization is driven by short range interaction (exchange of resource). The paper is organized as follows, in Sec. 2 we discuss the ingredients of model inspired from trees, trying to keep a minimal set of ingredients or variables. Then in the following section 3, we propose a way and implement the self-consistent dynamics of the growth of the tree. In Sec. 4 and Sec. 5 we perform some analytical calculations and predictions of the model and then perform some more analysis with numerical simulations in Sec. 6 before concluding in Sec. 7.

2 A bio-inspired model

In this section we construct a “growing” system centered around resource distribution and allocation inspired by biological trees. Considering how complex trees are, we opted for what appeared to be a simpler description of their growth and resource allocation. The main constituent of a tree is carbon, it is absorbed by their leaves and their source of energy is sunlight which also involve leaves. Both the carbon and energy is distributed and allocated in the form of sugars through the phloem. So for a simplistic description of trees we may shave off the xylem, water transport and the roots, which leaves us with a model featuring leaves as our sources, sugars as our resource, branches representing the nodes of our growing transportation network, and the volume of carbon each branch has fixated. Each branch will need to regularly consume the resource (sugar) to increase in volume (of carbon) and to keep its cells functioning. Let us introduce some terminology for our model:

• •

Trees are modeled as acyclic (i.e. loopless) connected networks like in figure 1, the nodes are called branches, one of them is designated as the trunk which is the only node existing at the start of the simulation.

• •

Each branch has a height which is an integer indicating its distance to the trunk (the trunk is of height 0, the branches directly linked to it are of height 1 etc). A branch with greater height than another is said to be higher.

• •

If two branches are linked by an edge then the highest one is said to be the child of the second. The second is the parent of the first. The terms ancestors/descendants will be used for parent/children, grandparent/grandchildren, etc. And we say a branch A descends from another one B when the branch A is a descendant of B.

• •

Finally, we call extremities branches without any children. And the height of the tree is defined as the height of the highest extremity.

Now, we propose a few rules related to resource distribution and allocation:

• •

The extremities will be the sources (the only branches possessing leaves) and all the branches, including the extremities, need to consume sugars as maintenance cost.

• •

Each branch can also *use up its resource to increase its *volume, the idea being large volume equates with more resilient branch, but in exchange the maintenance cost is higher for a larger branch.

• •

Each branch can transfer resource to any of its children or to its parent and it can also store it in its “personal” reserve.

And the rules driving changes in the network are the following:

• •

If a branch dies (could not pay its maintenance cost) then all its descendants will die (the branch is cut off the tree). Since the sources are at the tips of the tree, it creates an interdependence between the ones producing resource and the rest of the tree.

• •

About growth: extremities can use up resource to create children and in exchange cease to be sources while the newborns will be the new extremities. It is through this process that the tree can increase its number of sources since each time a branch creates more than one child, the number of sources increases.

Time is discrete: the branches act in successive rounds. We will provide later an example of how this “game” may progress.

The last characteristic we need to detail is the volume of branches. Each branch possess their own quantity of volume (i.e. they have a certain size). The bigger a branch is the more costly its maintenance is, and to become bigger the branch also needs to spend resources. In our model, we decide that the minimum volume (in some arbitrary unit of volume) a branch needs to have is equal to the number of its descendants that are extremities: if a branch has one child that is an extremity, two grandchildren that are extremities and one great grandchild that is an extremity, then the branch at least needs a volume equal to 4 (in unit of volume). This rule will be called Leonardo’s rule because it is inspired from the real Leonardo’s rule which has been studied in plantsleonardo_dav ; SHINOZAKI1964KJ00001775191 ; MCMAHON1976443 ; PhysRevLett.107.258101 .

The exact formulation of the real Leonardo’s rule is a rule of conservation of cross section areas: when looking at a branch that branches out into several others ones, the cross section of the branch before the branching node is equal to the sum of the cross sections of the branches after the node. Our Leonardo’s rule would be synonymous with the real one if consider the branches in our model as cylinder with all the same length and the extremities of our tree all have a volume of 1 unit.

3 Example of implementation

We have established the basic rules describing our model. However such description does not specify the model we implement. We still have to describe how the “game” progresses in time or, in other words, the dynamics of the system.

Example of a possible dynamics: we start with an amount of resource created at each source then during each “turn” every branches can store, consume for volume growth or transfer any amount of resource to their neighbors. And then after $x$ number of “turns” each branch has to pay maintenance from their stored resource or die and we repeat process. Creating new branches may involve consuming resource for $y$ numbers of unit of time etc.

Such dynamics would be fairly difficult to study, so we opted for a simpler dynamics. Unlike the example described above, in our chosen version, the branches do not act simultaneously but in succession. We will call this succession of events and actions a generation. When a generation ends, a new one begins, so generations will be used as our unit of time. Each generation is composed of two phases and each branch acts once during each phase. The first phase describes how the amount of resource generated at the extremities flows “downward”: from the extremal branches to eventually the trunk. And the second phase describes how the resource reaching the trunk bounces back “upward”: from the trunk to the extremities.

Let us specify this first phase which we call “flux down” (Fig. 2): the first branches to act are the extremities, then the next to act are the branches for which all the children have already acted, and we repeat until every branches have acted. Each *action *is the following sequence of events:

• •

If the branch is an extremity then it receives an amount $p_{0}$ of resource. Otherwise, it receives an amount from its children and will remember which amount was received from which child.

• •

It also receives from each child information about their energetic needs which is their maintenance cost as well as the resource they need to create the additional volume of wood necessary to support the structure above them. This information about energetic needs determines how the resource will be distributed during the second phase when the parent will have to choose the amount to distribute to each child.

• •

The branch can keep some of the resource in its reserve then transfer the rest to its parent. Then, it is this parent that will perform this same sequence of events.

In the second phase which we call “flux up”, the branches also act in succession but the order is reversed (Fig. 2) and the actions are now:

• •

If the branch is the trunk then it will start with all the resource that was given to it during flux down, then the flux will bounce back. If it is not the trunk it will have the resource it kept in its reserve but has also received an amount from its parent.

• •

With this amount of resource, it must pay its maintenance cost which depends of its volume and dies if it can not.

• •

Then, it will try to use up resource to grow to its “optimal” volume which is equal to the number of extremities that are its descendants (the volume of a newborn branch being set to one unit of volume, it means if the volume of each branch is “optimal”, in the sense we have defined above, then the tree will perfectly respect Leonardo’s rule). If the branch volume is not larger or equal to its optimal volume, it is cut off.

• •

The branch shares all its remaining resource to its children. The share given to each one depends on what happened during the flux down phase: the energetic needs each one told as well as the amount of resource each gave.

• •

If the branch has no child (it is a source) then it uses the resource to create children. There is a maximum number of children it can create.

In this phase, the proportion a parent shares/distributes its resource to one child or another is decided by a formula of our choosing. However the parent is supposed to remember the amount of resource each child gave during flux down as well as their energetic need. So a suitable formula would be one that use these two values, doing so we obtain the tree represented in Fig. 3.

In the end the dynamics of the tree only depends on a few parameters:

The quantity produced by one extremity $p_{0}$. 2. 2.

The cost of creation of a new branch $\mathcal{C}$. 3. 3.

The maintenance cost of a branch of volume $V$: $m_{0}\times V{}^{\alpha}$. 4. 4.

Cost of creating more volume of wood (increasing the branch’s diameter). The cost is linear $C_{v}\times(\textrm{Vol created})$. 5. 5.

The maximal number of children a branch can create $N_{max}$.

Using the simple dynamics we described, we can obtain the tree that was represented drawn in three dimensions in Fig. 3. All the branches were identical: during flux down, they all use the incoming resource using the same algorithm (they grow in volume in order to get the volume dictated by Leonardo’s rule) during flux up, they all use the same algorithm to decide how they share their resource among the children (the algorithm take as entries the energetic need of a child and its previous contribution during flux down) but with a very small proportion of the resource being sometimes distributed randomly. As for the branches with no child, they would use the remaining resource to create as many children as possible.

On the other hand, if, during flux up, we do not implement that slight random distribution then the branches will always split equally their resource among their children because every branch have the same strategy so, during flux up redistribution of resource to the children, the parent will have no reason to discriminate among the children. This results in “symmetric” trees. Either way, the growth and trees we obtain depend on the parameters ($p_{0}$, $\mathcal{C}$, $m_{0}$, $C_{v}$, $\alpha$ and $N_{max}$). Therefore, we may want to have a theoretical view on the system. The simplest way to look at the system is by having a static approach: for instance, for a tree with a given topology we can calculate the total resource produced and its total maintenance cost assuming the volume of the branches follows Leonardo’s rule.

4 A static description of the problem for the case $\alpha=1$

If we solely look at the total production and maintenance cost of a tree for a given topology, it will allow us to make statements that are independent of the dynamics we chose, as such, this is the approach we will start with. For the calculations in this Section, the only hypothesis that will be used are: every extremities produce the fixed amount $p_{0}$ per unit of time, each branch pays the cost $m_{0}\times(\textrm{Vol branch})^{\alpha}$ per unit of time and the volume of the branches are such that they respect Leonardo’s rule, i.e no branches have been cut above it.

We may want to interpret $\alpha$. If we were look at our tree in the context of botany: the branches in our model are supposed to be cylinders of same length and the only morphological difference between the branches is their radius, therefore the volume scales like its cross-section surface. $\alpha=1$ means the maintenance cost is $m_{0}\times(\textrm{Vol branch})$: this proportionality to the volume can be interpreted as the cells being uniformly spread in the branch and consuming resource at the same rate. In reality, for large trees and branches, living cells are thinly located at the exterior of the branch while the interior of the branch is mostly dead wood: this would correspond to $\alpha\simeq 1/2$. Outside the context of botany an analogy to be drawn between our cost maintaining a structure (volume) and the results on dissipative cost of transporting resource found in PhysRevLett.84.4745 . (The comparison mainly holds thanks to the conservation of volume i.e. Leonardo’s rule.) Their paper found trees (loopless network) as the “optimal” topology when $0<\alpha<1$ but not in other cases. Considering this fact, we will ignore $\alpha>1$. The case limit $\alpha=1$, however, can still give us some insight for $0<\alpha<1$ and makes analytical calculations easier.

Since we assume all extremities (and only them) create an amount of resource $p_{0}$, then the total production, noted $\mathcal{P}$, is equal to $p_{0}\times E$ where $E$ is the total number of extremities the tree has. Now, we recall that, for each branch, its height is an integer defined as the distance between the branch and the trunk within the graph representing the tree. For each extremity of the tree, we can note its height, then we define $E_{h}$ as the number of extremities that have a height equal to $h$, as a result $E=\sum E_{h}$. Thus, by defining $H$ as the height of the highest extremity, we can write:

[TABLE]

We call $\mathcal{M}$ the total maintenance cost. It has a simple expression when $\alpha=1$ (the linear case):

[TABLE]

To illustrate this formula we can look back at Fig. 1: if Leonardo’s rule is respected the trunk has a volume of 5 because there are 5 extremities that are its “descendants”. Again with Leonardo’s rule, if we look at the children of the trunk (the branches named A and B): one has a volume equal to 2 and the other to 3. In other words, summing the volume of the children of A and B also yields 5 which is the number of extremities that either “descends” from A or from B. Then if we look the branches at the height just above, the sum is also 5 which is the number of extremities descending from them plus the number of extremities among them. This last reasoning yields 2 for the branches of height 3.

This observation can be generalized into the following rule: for a tree of height $H$ and given any integer $j\in[0,H]$, the sum of the volume of all branches that have a distance $j$ from the trunk (height equal to $j$) is equal to $A_{j}=\sum_{h=j}^{H}E_{h}$. So the total of volume of the tree is $\sum_{j=0}^{H}A_{j}$ which can be rewritten as $\sum_{h=0}^{H}(h+1)\times E_{h}$. A way to interpret this last expression is by saying that adding an extremity of height $h$ increases the total volume by $(h+1)$ units. Thus, if the maintenance cost of a branch is proportional to its volume (i.e. $\alpha=1$) then, by linearity, we get the equation (2) by multiplying the total volume of the tree with the proportionality factor $m_{0}$. Hence for $\alpha=1$, by combining (1) and (2) we obtain the global balance $\mathcal{B}$:

[TABLE]

This expression for $\mathcal{B}$ is simple enough we can easily deduce a few results. We recall $N_{max}$ is the maximum number of children a branch may have. For a fixed $N_{max}<\infty$, and using the expression (3), we obtain the following results:

We can not find arbitrarily tall tree such that $\mathcal{B}\geq 0$. This in turn implies that no tree can grow infinitely in height as they would reach a height above which $\mathcal{B}$ can only ever be strictly negative. 2. 2.

For trees of height $H\leq H_{0}$, with $H_{0}$ being an integer we will define later in Eq. (4), we know the form of the tree maximizing $\mathcal{B}\geq 0$: this maximal tree is the tree for which every branches have $N_{max}$ children, except the branches of height $H$. I.e. this is the tree of height $H$ that has $N_{max}^{H}$, hence maximizing its number of extremities.

In order to show this we start by defining the integer $h_{max}=\left[\frac{p_{0}}{m_{0}}\right]-1$. The brackets $\left[x\right]$, indicate the integer part of $x$, and will be used as such hereafter. This is the smallest integer for $h$ such that $p_{0}-(h+1)m_{0}$ is negative for any $h>h_{max}$. Therefore, in Eq. (3), the terms of indexes higher than $h_{max}$ are negative. In other words, any extremities added at a height strictly higher than $H_{max}$ will contribute negatively to our profits $\mathcal{B}$. Consequently, since $N_{max}<\infty$, it is clear we can not find arbitrarily tall trees such that $\mathcal{B}$ as we would reach a point where adding branches will only contribute negatively.

Now, we prove the second result: let us consider a tree. We consider a branch such that its height is strictly less than $H_{max}$ and has a number of children $n_{c}$ such that $1<n_{c}<N_{max}$ (so we precisely pick a branch that is not an extremity). If we build a tree identical to this first one except that the aforementioned branch is given new children until it has $N_{max}$ children (with the added children being extremities (i.e. have no child) ) then the second tree will have a higher $\mathcal{B}$ than the first one. An illustration of this statement is in Fig. (4). The second step is to take, this time, an extremity and to give it $N$ children which will become the new extremities, then we can look at how $\mathcal{B}$ changes. Since we now have $N-1$ more extremities, the production increases by $(N-1)p_{0}$ but the maintenance cost increases by $((N-1)(h+1)+N)m_{0}$. With a bit of algebra we can easily determine the values of $h$ for which the increase in production compensates the increase in maintenance: if the height of the extremity we are giving new children is equal or lower than $h_{0}=\left[\frac{p_{0}}{m_{0}}-\frac{N}{N-1}\right]-1$, then the contribution to $\mathcal{B}$ is positive. It is negative otherwise, and leads to

[TABLE]

By combining the two previous remarks, we conclude that, below a certain height, the tree with the highest $\mathcal{B}$ is the tree with $N_{max}^{H}$ extremities and, beyond this height, any adding new children to a extremity lowers $\mathcal{B}$.

5 Generalization for $0<\alpha<1$ and infinite trees

To sum up, the two important facts about $\alpha=1$ we have established are: no infinitely growing tree is viable and creating as much children as possible yield the most productive trees as long as the height of the tree is under a certain height $H_{0}$.

Now, for $0<\alpha<1$, we may ask whether arbitrarily tall trees that are viable exist. In other words, we want to know whether, for each height $H$, there exists one tree such that $\mathcal{B}>0$ and what constraints on $p_{0}$ and $m_{0}$ are needed. Obviously, calculating $\mathcal{B}$ for all possible trees of height $H$ may not be the best strategy. However, inspired from the results of the $\alpha=1$ case, we can first limit our search to trees of height $H$ with $N_{max}^{H}$ extremities (trees maximizing its number of branches). To illustrate how $\mathcal{B}$ can be calculated, let us take the example$N_{max}=3$ and $H=3$ then we can easily count that the trunk has a volume of 27, its 3 children have a volume of 9, its 9 grandchildren have a volume of 3 and, at last, there are 27 extremities. Therefore, the total maintenance is:

[TABLE]

Using the same kind of reasoning for a more general $N_{max}$ (that we will note as $N$ from now on) and any $H$, we can deduce the Eq.␣(6):

[TABLE]

From there we have an expression for $\mathcal{B}_{H}$ when $N>1$ and $\alpha\neq 1$ :

[TABLE]

From this expression we can see that if $0<\alpha<1$, then $p_{0}-m_{0}\frac{N^{1-\alpha}-N^{-(1-\alpha)H}}{N^{1-\alpha}-1}\geq p_{0}-m_{0}\frac{N^{1-\alpha}}{N^{1-\alpha}-1}$ for every $H$ therefore, for a given $N$ and $\alpha$, the value of $p_{0}/m_{0}$ determines whether $\mathcal{B}_{H}$ will eventually become negative. So, we have the following two cases:

[TABLE]

In the first case, $\mathcal{B}_{H}\rightarrow\infty$ and, in the second case, $\mathcal{B}_{H}$ becomes strictly decreasing after a certain $H$, then $\mathcal{B}_{H}\rightarrow-\infty$. So, a sufficient condition for the existence of viable trees (i.e. $\mathcal{B}>0$) at any height is $p_{0}/m_{0}\geq p_{critical}$.

This condition is also necessary, but we need to go through multiple arguments to reach this conclusion. A first observation to make is that branches have no benefit having only one child since only having one do not increase the number of extremities while increasing the maintenance. So, we will only discuss trees for which every branches, except the extremities, have at least 2 children. With this kind of trees, for any $m\in\mathbb{N}$, we can find $H_{m}\in\mathbb{N}$ such that any trees of height $H_{m}$ or higher have $m$ or more extremities.

We make a second observation: let $m\in\mathbb{N}$, we can find $H\in\mathbb{N}$ such that $N^{H}\leq m<N^{H+1}$. If we construct a tree with $m$ extremities by starting from the tree of height $H$ that has $N^{H}$ extremities, then adding the remaining $r_{0}=m-N^{H}$ new extremities in the way described in Fig. (5), this tree has a maintenance cost lower or equal than any other trees with $m$ extremities. Let us verify this last statement. First, it is obvious we there can not exist a tree with $m$ extremities that is strictly shorter than the one we just constructed because no branch may have more than $N$ children. Second, let us consider strictly taller trees with $m$ extremities: they would have more total volume (since we recall the formula for the total volume is $\sum_{h}(h+1)E_{h}$) and share this total volume among more branches, therefore, using the inequality $(\sum x)^{\alpha}\leq\sum x^{\alpha}$, we deduce their maintenance cost must be higher. Third, now we simply need to compare trees which height is $H+1$: they all look like our tree except that the $r_{0}$ remaining extremities may be split differently. Our splitting minimizes maintenance costs because $v^{\alpha}+(v+x_{1}+x_{2})^{\alpha}\leq(v+x_{1})^{\alpha}+(v+x_{2})^{\alpha}$ which means it is better to pack the $r_{0}$ branches together in one side.

A third and last observation has to be made. We mentioned that when the condition is not respected (i.e. $p_{0}/m_{0}<p_{critical}$), $(\mathcal{B}_{H})_{H}$ is strictly decreasing and negative after some $H_{f}$. We can establish that if $H\geq H_{f}$, then the tree with $m$ (such that $N^{H}\leq m<N^{H+1}$) extremities described earlier (Fig. (5)) must have a $\mathcal{B}$ inferior to $\mathcal{B}_{H}$ (therefore negative too). Indeed, $\mathcal{B}$ not being inferior to $\mathcal{B}_{H}$ iscontradictory: we name again $r_{0}=m-N^{H}$ and assume $r_{0}=N-1$ then conclude that if the resulting $\mathcal{B}$ is superior to $\mathcal{B}_{H}$, then $\mathcal{B}_{H+1}\geq\mathcal{B}_{H}$ because in this case adding one (or more) group of $N-1$ extremities will increase $\mathcal{B}$ even more. Consequently, the case $r_{0}=N-1$ must lowers $\mathcal{B}$ which we can then use to prove the case $r_{0}=k(N-1)$ also lowers it, and after this we deduce it for a more general $r_{0}$.

Finally, using the first observation, we can find $H_{1}$ such that every trees of height higher than $H_{1}$ have at least $N^{H_{0}}$ extremities. Then, for each of these tree, using the second observation, we can create a tree, similar to the one in Fig. (5) with the same number of extremities but with a lower $\mathcal{B}$. But, this last tree has a height $H\geq H_{0}$ so its $\mathcal{B}$ is negative, using the third observation. Thus, when $p_{0}/m_{0}<p_{critical}$, arbitrarily tall and viable trees do not exist.

Now, let us go back to Eq. (7) and contextualize it. Until now, we looked at the static cost of the branches. By doing so we ignored an important aspect of our dynamical tree: branches need to actually spend resource to grow into the volume of Leonardo’s rule; each time a branch increases in volume it consumes an amount proportional to this volume increase (Sec. 3: $C_{v}$ being the proportionality coefficient). Let us look at the cost induced by this process when we go from the tree of height $H-1$ and $N^{H-1}$ extremities to the tree of height $H$ with $N^{H}$ extremities.

[TABLE]

If we can compare $\mathcal{C}_{vol(H-1\rightarrow H)}$ to $\mathcal{B}_{H}$, it is $\mathcal{C}_{vol(H-1\rightarrow H)}$ that dominates for large $H$ ($\mathcal{B}-\mathcal{C}_{vol}$ tends to $-\infty$ as $H$ grows). However unlike a maintenance cost, this creation cost is only paid once each time the tree grows, so a tree that takes time to accumulate a reserve between each growth spurt may grow arbitrarily tall. But the amount of time waiting to accumulate a reserve will increase as $\mathcal{O}(H)$. Simply put, if $C_{v}\neq 0$ then our tree will not be able to grow infinitely tall unless it is “intelligent” enough to slow down its growth, and even then we predict a “rate of slow down” that is exponential. It may echo the fact real trees have their growth slowing down with sizeMencuccini2005 .

6 Simulation of symmetric trees

6.1 Encoding the trees in sequences of integer

Up until now, all our theoretical results are independent of the dynamical rules we implement: they all are simple arguments on topology and economic costs. Let us go back to the dynamical system described in section 3, and we consider the case where all the branches has the same strategies during flux up (the phase in which the flow go from the trunk to the extremities): at the end of each generation, the extremities generate as much children as possible with the resource they possess. As explained in Sec. 3: without asymmetry or randomness, the tree will be “symmetric”, meaning that all branches located at the same height will have the same number of children and brothers, the same volume etc. The trunk will have $u_{0}\leq N$ number of children. By symmetry, these $u_{0}$ children will all have the same number of offspring, noted $u_{1}\leq N$, etc. So, for a tree of height $H$, we can construct the sequence $u_{0}$,$u_{1}$,…,$u_{H-1}$ where $u_{i}\leq N$. A tree composed of only a trunk is represented by an empty sequence. Conversely such a sequence will define a symmetric tree of height $H$.

Hence we can see the growth of our simulated symmetric trees as a finite sequence of $u_{i}\leq N$ that evolves dynamically. While the ”form” of a symmetric tree of height $H$ is characterized by a sequence $u_{0}$,…,$u_{H-1}$, the tree itself is also defined through how the volume and the reserve is allocated among the branches. Since every branches located at the same height have the same volume and reserve then the full characterization of a tree of total height $H$ is given by 3 finite sequences: $(u_{i})_{0\leq i\leq H-1}$, $(v_{i})_{0\leq i\leq H}$ representing the volume $v_{i}$ of the branches located at height $i$ and $(r_{i})_{0\leq i\leq H}$ giving the amount of reserve $r_{i}$ stored in the branches located at height $i$. With our symmetrical dynamics the sequence $(u_{i})_{0\leq i\leq H-1}$ may shrink (branches of height superior to some $h$ die) or may grow (new integers $u_{H}$, $u_{H+1}$ etc added) but the values $u_{i}$ themselves do not change because if a branch of height $h+1$ dies, those of the same height also all die by symmetry, therefore $u_{h}$ would simply disappear from the sequence as well as all $u_{i>h}$.

The initial condition for all our simulation is $(u_{i})=\varnothing$ (symbol for empty sequence), $(v_{i})=(1)$ and $(r_{i})=(0)$ (i.e. we always start with a tree formed only by a trunk of volume 1 with no initial reserve). Given how, we kill every branch that has a volume less than what Leonardo’s rule would predict then, as already mentioned, the only time some $v_{i}$ do not follow Leonardo’s rule is when a tree got some branches dead (so that the survivors will have bigger volume than predicted by Leonardo’s rule). So if we interest ourselves with infinitely and steadily growing trees the sequence $(v_{i})_{i}$ is unneeded since it matches Leonardo’s rule perfectly. Though, $C_{v}=0$ is a requirement if we want trees growing ever steadily.

6.2 Evolution of $(u_{i})_{i}$ for infinitely and steadily growing trees

We define a “steadily” growing tree as a tree such that at the end of generation number $i$, the height of the tree is also $i$ (i.e. the tree never shrinks and never stagnates). We have stated at the end of Sec. 5 that unless $C_{v}=0$ an infinitely and “steadily” growing tree is impossible, so from now on $C_{v}$ is taken as 0 (i.e. growing in diameter/volume do not cost any resource to the branches). If we note $B_{H}=u_{0}\times u_{1}\times...\times u_{H-1}$ and define $B_{0}=1$ then the $B_{i}$ represent the total number of branches located at the height $i$. If we want to write an expression for $u_{H+1}$ or simulate the system, we first need to place ourselves in a specific version of the system described in Sec. 3. For the specific system we study: during “flux down”, the children give everything to their parent which means that at the beginning of “flux up” the trunk start with the total production. Then, during “flux up”, the parents only pay their maintenance, are eliminated if they can not pay and then grow in volume (which is free since $C_{v}=0$) but do not keep anything more in reserve and distribute equally to their children the remaining flux. Finally when the flux reaches the extremities, they, after paying their maintenance, use as much resource as possible to create as many children as they can (the cost of creation for each child being an integer $\mathcal{C}$). With this choice of evolution rule, the system is greatly simplified because, right before the extremities have to create new children and at the end of all the maintenance payment, the resource the extremities have is equal to the total amount produced (by the extremities at the beginning of “flux down”) minus the total maintenance cost of the tree plus the leftover resource that could not be transformed into children the turn before, noted $R_{i}$. So since we know the resource the extremities have right before creating children, we can predict the number of children each extremity will spawn, $u_{H+1}$:

[TABLE]

Of course, if there is a limit $N_{max}<\infty$, we take the minimum between the expression above and $N_{max}$.

$R_{H+1}$ is the unused leftover during the previous cycle in other words:

[TABLE]

The big first term in the numerator of Eq. (10) is the total production minus the total maintenance. And at the denominator, we take into account the fact the $B_{H+1}$ extremities share this total resource and $\mathcal{C}$ is the cost of a child. This formula only works as long as the tree grows “steadily”, meaning that if along the flux up some branches could not pay maintenance or if the extremities do not have enough to create a child ( $u_{H+1}=0$ ), then (10) and (11) stop being predictive. Conversely, as long as all the $u_{H+1}$ give strictly positive integer, we can be sure no branches died along “flux up” (otherwise the numerator in (10) would be negative and so would $u_{H+1}$). In other words, the moment $u_{H+1}\leq 0$ is the moment the tree has stopped growing “steadily” and we note $H_{f}$ the last height/generation (10) is valid. $H_{f}$ may or may not be infinite but we are particularly interested in the infinite case.

6.2.1 The case $R_{i}=0$: the tree does not keep its leftover resource

Now we put the term $R_{H+1}$ to zero and since $u_{H}=B_{H+1}/B_{H}$, we obtain a simple expression to get the successive $B_{H+1}$.

[TABLE]

To simplify things we introduce $W_{n}={\displaystyle\sum_{i=0}^{n}(B_{i}/B_{n})^{1-\alpha}}$.

[TABLE]

Then we replace the $B_{n+1}$ term at the denominator by the expression in (12) which replaces $B_{n+1}$ by $B_{n}$ multiplied by a factor. This term $B_{n}$ allows us to recover a $W_{n}$, and we end up with a “simple” recurrence.

[TABLE]

Just as with Eq. (10), this expression is valid only until $n=H_{f}$ which corresponds to the first integer that yields $\left\lfloor a_{0}-W_{H_{f}}\cdot b_{0}\right\rfloor\leq 0$. Since we are interested in trees that can grow infinitely we limit our study to $0<\alpha<1$, and in this case $H_{f}$ may be infinite for some values of $a_{0}$ and $b_{0}$. $H_{f}=\infty$ is equivalent to $W_{n}$ (equivalently $B_{n}$) being defined for any natural number $n$. And if these two sequences are defined for every $n$ we may want to look if they converge or not. First we need to establish when do we get an infinite sequence ( $H_{f}=\infty$ ). We also remind $a_{0}$, b_{0}$$>0. $p_{0}>m_{0}$ is also assumed because otherwise our tree will stop growing at $H_{f}=1$.

We define the function $f_{0}$ such that Eq. (13) simply becomes a discrete dynamical system $W_{n+1}=f_{0}(W_{n})$ and $f_{1}(x)=1+x/\left(a_{0}-x\cdot b_{0}\right)^{1-\alpha}$, that we can study with classical tools. In fact $f_{1}$ is a simplified and a more easy-to-study version of $f_{\text{0}}$. Their behavior with respect to $\mathbb{I}:x\rightarrow x$ provides us with the behavior of $(W_{n})$.

The figure (6) provides us with the behavior of these functions with respect to one another. Before moving to study this dynamical system, we point out a few things about $f_{0}$, $f_{1}$ and $W_{n}$:

• •

The domains of definition of our functions are $\mathcal{D}(f_{1})=(-\infty,\textrm{ }a_{0}/b_{0})$ and $\mathcal{D}(f_{0})=(-\infty,\textrm{ }(a_{0}-1)/b_{0}]$. The functions being positive and the initial condition being $W_{0}=1$, only $\mathbb{R}^{+}\cap\mathcal{D}(f_{1})$ and $\mathbb{R}^{+}\cap\mathcal{D}(f_{0})$ interest us. The sequence $(W_{n})_{n}$ stops when $f_{0}(W_{n})$ escapes the domain of definition of $f_{0}$ and the term $H_{f}$ is the last defined one.

• •

Given $a_{0},$ $b_{0}>0$ and $0<\alpha<1$, both $f_{0}$ and $f_{1}$ are strictly increasing in the domains we are interested in.

Then we can show that even for a non-continuous function like $f_{0}$, we end up with an attractor which is a simple fixed point. (see Appendix Appendix A for details) this means that the sequence $W_{n}$converge in most cases, it is indeed possible that $f_{0}$ is not below the line $y=x$.

Now let us suppose we do have a fixed point. We call this fixed point/ limit $W^{*}$. But

[TABLE]

so $(u_{n})_{n}$ converges too. If we assume $N_{max}>\left\lfloor a_{0}-W^{*}b_{0}\right\rfloor$ then $u^{*}=\lfloor a_{0}-W^{*}b_{0}\rfloor$. So if we fix $a_{0}$ and $b_{0}$, the limits of both $(W_{n})$ and $(u_{n})_{n}$ will be tied together. The conclusion, which is observable through the simulations, is that when we negate the reserve $R=0$ then infinite trees give sequences $(u_{i})$ that ends with an infinite succession of $u^{*}$.

In conclusion, with the dynamics and strategy we described (there is no external threat and the branches are acting brainlessly), infinite trees only appear for some values of the parameters and they end up, after a certain time, growing in a very regular way: the number of extremities is multiplied by $u^{*}$ each generation and we end up with a self-similar tree.

6.2.2 Case with the reserve on: back to equation (10).

First we can make the following remark: even with the reserve turned on, the results about the necessary conditions to get an infinitely growing tree described earlier should, at least to some extent, hold true. However with the addition of the reserve, we observe (from numerical calculations) that instead of having a sequence $(u_{i})_{i}$ that ends with an infinite succession of $u^{*}$, we may sometimes have a periodicity: the sequence will end up oscillating between $u_{1}^{*}=\lfloor a_{0}-W^{*}b_{0}\rfloor$ and $u_{2}^{*}=\lceil a_{0}-W^{*}b_{0}\rceil$. There does not seem to be limits for the length of the periods we find. But, the period we get, vary progressively in function of the parameters $a_{0}$ and $b_{0}$ (Fig. 8). So while we have a less repetitive growth than in the case $R=0$, the tree grows in a very regular manner and do not seem to change erratically its growth pattern when we vary slowly the parameter values.

7 Conclusion and perspectives

In this paper, we have presented a set of basic rules describing a group of bio-inspired dynamical systems focused on resource distribution and allocation. By completing these basic rules with specific evolution rules, we construct a growing tree-like system we can simulate and study. However, even with only the basic rules and no evolution rules, we have shown it was possible to get a few results about the possible topologies of a tree-network that possesses sources at its extremities and have a maintenance cost for each node which increases the closer the node is to the root (Leonardo’s rule). Then, we specified the evolution rules and studied the resulting dynamical system. By studying a very simple version of the system, we could established a few theoretical and numerical results that can be useful to lay some ground for future works. We emphasize as well that during our study we initially spend a lot of time scanning parameter space for the case $\alpha=1$, in this situation, as already discussed, most trees end up dying, however for some carefully chosen parameters we can end up with what would be an like an unstable fixed point, a tree that simply stops growing and lives forever. Nothing really interesting came out of this thorough numeric study which is why we did not present it in this paper, however these unstable tree allowed us to test the consequence of how the redistribution of resources among children affects the tree. The results are displayed in Fig. 9, we start from a perturbation of an unstable tree by adding three extra branches, this will lead to a depletion of the reserves and a death of the tree, but as illustrated we can see that depending on how the redistribution is made, the tree can be more resilient depending on the redistribution chosen . The formula that dictates the proportion a branch/parent will distribute its resource to each child depends on two elements: 1) the need of the child which is composed of its maintenance cost plus the cost of growing to reach the required volume (Leonardo’s rule) 2) the amount of resource the child gave to the parent during “flux down”. More specifically, if we note $e_{i}$ the need of a child $i$ and $c_{i}$ its contribution during flux down then the formula is $Ze_{i}^{r_{1}}c_{i}^{r_{2}}$. With $r_{1}$ and $r_{2}$ parameters and $Z$ the normalization constant ${\textstyle\sum_{j}}e_{j}^{r_{1}}c_{j}^{r_{2}}$ (sum over all the children $j$ of this particular parent), and thus if $r_{1}>r_{2}$ the tree redistribute resources based on needs, while the $r_{1}<r_{2}$ is based more on reward. The results displayed in Fig. 9 clearly show that for the considered case, a strategy based on reward keeps the tree alive, while the one based on needs ends up killing the tree. This leaves us with the perspective of this work: we dubbed this paper part 1 kinematics, as no external forces or interaction between the branches besides redistribution of the resources was taken into account, so only the self-sustained kinematics of the tree were taken into account. As a first perspective we want to embed this growth into real space, which will add occupation constraints on the new branches and some exclusion rules of available space to grow new offspring. This will lead to some interactions between the branches and we expect that the resulting dynamics will be greatly affected by this. Another aspect of future work, will be to influence the role of the redistribution parameter and its possible influence on the resilience of the tree structure as well as its overall shape like its extremities (foliage) when embedding it in real space. A comparison would be then possible with the already existing a attempts to obtain realistic looking tree shape from relatively simple rules Eloy2017 ; Duchemin2018 . Thus, with the embedding in space, we could examine whether our approach with resource distribution and allocation could yield similar results.

Ackowledgements

X.L. thanks Christophe Eloy, for fruitful discussions and contributions especially in the early development of the model, and its numerical implementation. O.B. and X.L. would like as well to thank Bruno Moulia, Eric Badel and André Lacointe for encouragements and useful suggestions. The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University - A*MIDEX, a French ?Investissements d?Avenir? programme. It has been carried out in the framework of the Labex MEC. We also acknowledge support from the CNRS (Mission pour l?interdisciplinarité, project ARBRE)

Appendix A

Let us show that even for a non-continuous function like $f_{0}$, we end up with an attractor which is a simple fixed point. Indeed, any real-valued function $g$ strictly increasing defined on $[x,\>y]$ such that $g(x)>x$ and $g(y)<y$, $g$ has a fixed point on $[x,\>y]$. Likewise with only these constraints on $g$ , if we define $u_{n+1}=g(u_{n})$ with $u_{0}=x$ as initial condition then the sequence is define for any $n$ and converges. We can apply this result to the function $f_{0}$ and the sequence $(W_{n})_{n}$ with $W_{0}=1$. First, we may want to set $a_{0}>b_{0}+1$ so that $W_{1}=f_{0}(1)$ is defined. And if it is defined then it is obvious $f_{0}(1)>1$. Therefore, the last step is to prove the existence of a point $z$ such that $f_{0}(z)<z$, so we can deduce that $(W_{n})_{n}$ is an infinite sequence and converges. On the other hand, if $f_{0}$ is always above the line $y=x+\epsilon$ then it is possible to show that $(W_{n})_{n}$ will terminate at some integer $H_{f}<\infty$. Since $f_{0}\geq f_{1}$ (cf. figures (6) and (7) ), studying $f_{1}$ can give us a sufficient condition for $H_{f}$ to be finite: having $f_{1}-\mathbb{I}>\epsilon$ would be that sufficient condition.

Let us study $f_{1}$ on $D=\mathbb{R}^{+}\cap\mathcal{D}(f_{1})$. Assuming again $a_{0}>b_{0}+1>1$ and $0<\alpha<1$, we have both $f_{0,1}(1)>1$. Now we want to see when would $f_{1}-\mathbb{I}>\epsilon$ or not i.e. when we can not or can find $y\in D$ such that $f_{1}(y)<y$. The derivative of $(f_{1}-\mathbb{I})$ is:

[TABLE]

Its sign is determined by its numerator that we will call $P(x)$; $P$ is defined on $\mathcal{D}(f_{1})$.

[TABLE]

So $P$ is, at first, strictly increasing until it reaches its maximum value at $x_{1}\equiv\frac{1}{b_{0}}\left[a_{0}-\left(\frac{\alpha}{2-\alpha}\right)^{\alpha}\right]$ then becomes strictly decreasing. In addition to that, because $2-\alpha>1$ we have $P(-\infty)<0$ and at the limit $x\rightarrow a_{0}/b_{0}$ we have $P(x)>0$. So we can now establish a variation table for $P$ and from there we deduce that there is a unique $x_{0}\in\mathcal{D}(f_{1})$ such that $P(x_{0})=0$ and $P(x)<0$ for $x<x_{0}$ while $P(x)>0$ for $x>x_{0}$. We can even specify a bit more the value of $x_{0}$: using the fact that $a_{0}>1$ we get $P(0)=a_{0}-a_{0}^{2-\alpha}<0$, furthermore $P(x_{1})>0$. The conclusion from these two facts is $x_{0}\in(0,\>x_{1})$. (Remark: $x_{1}>\frac{a_{0}-1}{b_{0}}$ because $\left(\frac{\alpha}{2-\alpha}\right)^{\alpha}<1$ and $a_{0}>b_{0}+1>1$.) From the sign of $P$, we finally deduce the variation table of $f_{1}-\mathbb{I}$. In conclusion, $f_{1}-\mathbb{I}$ reaches its minimum at $x_{0}\in(0,\>x_{1})$; $x_{0}=\frac{a_{0}}{b_{0}}-\frac{X_{0}}{b_{0}}$ where $X_{0}$ is the (unique) root in $\mathbb{R}^{+}$ of $a_{0}(1-\alpha)+\alpha X-X^{2-\alpha}$. If $f_{1}(x_{0})-x_{0}=\epsilon$ with $\epsilon>0$ then $H_{f}$ is finite since it would imply $f_{1}-\mathbb{I}\geq\epsilon$ as $x_{0}$ was the minimum of $f_{1}-\mathbb{I}$. So $f_{1}(x_{0})-x_{0}=\epsilon$ is the sufficient (and probably nearly necessary) condition for $H_{f}<\infty$ and it only requires a numerical determination of $X_{0}$ then a calculation of $f_{1}(x_{0})=1+x_{0}/X_{0}$.

Up to now, we saw that $f_{0}$ being below the line $y=x$ for some real in $\mathcal{D}(f_{0})$ implies the existence of a fixed point and $H_{f}=\infty$. Then we have managed to find some sufficient condition for it not to be the case. Now let us suppose we do have a fixed point. Since $f_{0}(1)>1$ the fixed points should be located after $1$. Considering how $f_{1}$ vary, it does not have more than 2 fixed points. We will assume the same for $f_{0}$; $(W_{n})_{n}$ will then converge toward the smallest fixed point of $f_{0}$ that we will assume to be equal to the smallest fixed point of $f_{1}$ (both should be very close from each other).