Invasive plants and animals may cause significant problems to natural environment, agricultural industries and amenities for example. In Victrola, Australia, a biosecurity approach has been in place to control invasive species (a risk management to identify threats and then assess their relative risk to determine the most appropriate intervention to maximise public benefit).  

The strategic biosecurity approach includes

(a) preventing the entry of new high risk invasive  plants and animals,

(b) eradicating of those invasive species that are at an early stage of establishment and

(c) containing where possible those species that are beyond eradication and

(d) taking an asset-based approach to managing widespread species.

Analyses of biosecurity programs reveals that prevention provides a higher return on investment than eradication, eradication is better than containment and containment is better than managing impacts of widespread invasive species.

In general, preventing high-risk invasive species from establishing is the most cost-effective approach to managing the threat they pose. Eradication is generally only possible in the early stages of establishment when distribution and abundance of the invasive species are low. Where a high-risk species cannot be eradicated, there can be substantial net benefit gained from preventing its further spread. Containment which involves measures to eradicate outlying (satellite) infestations and prevent spread beyond the boundaries of core infestations (those that are too large and well established to eradicate). An asset-based approach to managing an invasive species is appropriate once it has become so widespread that it would be inefficient to control the species everywhere it occurs and containment would provide a low return on investment. Lastly, invasive species management, monitoring, evaluation and reporting (MER) would be an important element of adaptive management

Reference

Invasive Plants and Animals Policy Framework. http://www.depi.vic.gov.au/agriculture-and-food/pests-diseases-and-weeds/protecting-victoria-from-pest-animals-and-weeds/invasive-plants-and-animals/invasive-plants-and-animals-policy-framework

Aloha Carlos,

In response to your question I think there are many examples form all over the world.  In Hawaii, fountain grass (Pennisetum setaceum), for example, is a highly invasive plant that has caused massive fire risks in many dry areas. People would love to eradicate it but the extent of itʻs spread and percent cover (often times the ONLY species covering huge areas) makes it nearly impossible.

However, I think that the concept you are looking at is becoming increasingly rare (at least in the US) due, primarily, to biological control.  In Hawaii strawberry guava (Psidium cattleianum) was battled for many years as an invasive plant to rare Hawaiian wet forests using physical and chemical controls. After some 20 years of fighting it on the ground those efforts have been mostly abandoned because the Strawberry guava was simply too aggressive. But a couple years later a biological control agent was introduced to reduce the fruiting potential. 

But many invasive species have been given up on their control in Hawaii, even in the most sensitive habitats.  The Kahili ginger (Hedychium gardnerianum) is another good example.  While in some select areas they still fight it with huge amounts of volunteer labor, overall the costs of fighting it are simply too high.

Thank you both, Golam and Noa, for your complementary answers!!

I know Pennisetum villosum in Catalonia (Spain), and Pennisetum setaceum is particularly problematic in southern Spain. In fact, some of the species discussed here are offered as elements of gardening.

Chromolaena odorata in many countries including South Africa.

Article Efficacy of Chromolaena odorata control in a South African c...

If you need more materials you may find this book:

The Great Reshuffling. Human Dimensions of Invasive Alien Species. Edited by Jeffrey A. McNeely. IUCN – The World Conservation Union. 2001.

good luck