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We study a particular first-order partial differential equation which arisen from a biologic model. We found that the solution semigroup of this partial differential equation is a frequently hypercyclic semigroup. Furthermore, we show that it satisfies the frequently hypercyclic criterion, and hence the solution semigroup is also a chaotic semigroup.

The first-order partial differential equations appear in different branches of science and succeed to demonstrate events of nature. In this paper we focus on the particular form of partial differential equation

This equation is usually used to represent the models of age-structured populations. Populations of replicating and maturing cells are age structured in that the replenishment of new individuals into the population depends on the density of a cohort of older individuals. Many biological populations have similar models; the Lasota equation is the famous example which is an application of (

Equation (

A lot of researchers are interested in the chaotic behavior of differential equation and chaotic

Motivated by Birkhoff's ergodic Theorem, Bayart and Grivaux introduced the notion of frequently hypercyclic operators in [

However, if a frequently hypercyclic semigroup

The arrangement of this paper as follows. we will find the solution semigroup

Using the method of characteristics to find the unique solution of problem (

For describing

Secondly, we are going to solve the initial value problem (

The characteristic of (

Let

Let

The space

We need only to show that

Since

For proving Lemma

There exists a closed subset

Let

For a given

For estimate

From (

From the properties of

For every

Using formula (

Pluging

At beginning of this section, we introduce some terminologies and propositions which will be used later. According to Devaney's definition, a semigroup

the set of periodic points of

We recall that the lower density of a measurable set

Let

According to this proposition one wants to show that a semigroup

Let

The proof of this proposition can be found in [

Suppose that

To show the conclusion of this theorem to be true, we are planning to apply Proposition

According to Proposition

For this purpose, we defined an operator

For checking condition (2) of Proposition

Let

According to the definitions of

From (

Using similar estimation of (

Although from frequently hypercyclic criterion we can get

From the definition of chaotic semigroup, we need to find the set of period points of

For any

It is not hard to prove that the set of periodic points of (

The solution

Finally, we demonstrate two simple examples. The first one is

where

It is easy to see that condition (

In fact, the solution semigroup

Another example is the Lasota equation (

The author would like to thank the referee for useful suggestions for this research work.