Introduction

Experiments  (link1, link2) indicate that plants do not grow exponentially but according a second-order polynomial. In an earlier paper (link3) is described how a n-order polynomial is derived from an exponential model. The essence of the reasoning is a shift from the growth machinery to more competitive ability. The question remained why a plant, growing according a second-order will win in a struggle for survival from those not growing according these characteristic.

To find an answer a computer simulation was made where plants growing according a second-order  were competing with slower growing (according a first-order polynomial) plant, see paper (link4) and with faster growing (according a third-order polynomial) plants, see paper (link5). But, it turned out to be important to evaluate a variable, which is unavoidable in the experiment: What should we use as maximum age?

Methods

See the earlier paper (link4).

Two simulations were conducted:

1. The first set of experiments started with only one plant. We did it for all growth characteristics: first, second and third order. It was allowed to grow and multiply to a high density. It means maximal competition. After a certain period we introduced with a mutant switch a change in maximum age. The switch was set off, when the amount of the mutants reached a percentage of approximately 30. So, the mutants have at that moment still a disadvantage in numbers.
2. The second set of simulations started with two plants with different maximum ages. We just looked if one would win. If one would win we restarted the competition with another one: If the  lowest maximum age would have won we choose an even lower one , and if the highest maximum age had won, we choose an even higher one. We did it again for all growth characteristics: first, second and third order.

Results and Discussion

(7-9-2020: Notice, the results were corrected as a mistake in the programming caused that only  the right side of the plant was competing. It was corrected without making a separate publication, because the main conclusions stayed the same. Only the maximum age of the concentrating second order polynomial plants changed significantly. This makes it interesting to investigate in a later stage if simulated plants with a one leaf on one side can compete with plants with two leaves, one on each side)

The first experiments showed no clear results in favour of a specific maximum age. It depended much on which age was in the majority at the start of the competition. The mutants lost most of the times, because they were in the minority when we set the switch off. Even when we interchanged the maximum ages this was the case.

In contrast, the second set of experiments showed an optimal maximum age, depending on growth characteristics, i.e. first, second or third order and concentrating or not concentrating in time and space, see table 1.

Table 1: Optimal maximum age for different growth characteristics

Order polynomial describing mass in relation with time	Concentration of offspring  in time and space
	no	yes
first	5	3
Second	no max age	30
third	no max age	8

It indicates that for concentrating simulated two-dimensional plants, an optimal death or maximal age, gives an advantage to conquer an empty limited space. This is is also the case for non-concentrating first order polynomial plants.  The outcome will maybe first a surprise, but there is an explanation: For a not  moving plant the death leads to an offspring and this offspring can spread around the space faster than the growing plant. But the mechanism is complicated: A later death means also more growth and a bigger offspring and killing more competitors.

The maximum age for the second order plants is, comparing to the other order polynomials, really striking . We will see what this means in another paper, link6.

We used the optimal maximum ages for first and third order plants to give them the greatest advantage possible over the second order plants. It makes the case stronger for the second order plants when they win, what they did when the offspring was concentrated in time and space.